JEE Main & Advanced Mathematics Vector Algebra Question Bank

done Vector or Cross product of two vectors and its application Total Questions - 88

• question_answer1)
  If a, b, c are any vectors, then the true statement is                                          [RPET 1988]
  A)
                  \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=(\mathbf{a}\times \mathbf{b})\times \mathbf{c}\]     done clear
  B)
                  \[\mathbf{a}\times \mathbf{b}=\mathbf{b}\times \mathbf{a}\] done clear
  C)
                  \[\mathbf{a}\,.\,(\mathbf{b}\times \mathbf{c})=\mathbf{a}\,.\,\mathbf{b}\times \mathbf{a}\,.\,\mathbf{c}\] done clear
  D)
                  \[\mathbf{a}\,.\,(\mathbf{b}-\mathbf{c})=\mathbf{a}\,.\,\mathbf{b}-\mathbf{a}\,.\,\mathbf{c}\] done clear
  View Solution play_arrow

• question_answer2)
  If a and b are unit vectors such that \[\mathbf{a}\times \mathbf{b}\] is also a unit vector, then the angle between a and b is
  A)
                  0 done clear
  B)
                  \[\frac{\pi }{3}\] done clear
  C)
                  \[\frac{\pi }{2}\] done clear
  D)
                  \[\pi \] done clear
  View Solution play_arrow

• question_answer3)
  The points \[A\,(\mathbf{a}),\,B\,(\mathbf{b}),\,C\,(\mathbf{c})\] will be collinear if
  A)
                  \[\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}\]     done clear
  B)
                  \[\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a}=\mathbf{0}\] done clear
  C)
                  \[\mathbf{a}\,.\,\mathbf{b}+\mathbf{b}\,.\,\mathbf{c}+\mathbf{c}\,.\,\mathbf{a}=0\]              done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer4)
  \[(\mathbf{a}-\mathbf{b})\times (\mathbf{a}+\mathbf{b})=\]                                      [MP PET 1987]
  A)
                  \[2(\,\mathbf{a}\times \mathbf{b})\]   done clear
  B)
                  \[\mathbf{a}\times \mathbf{b}\] done clear
  C)
                  \[{{a}^{2}}-{{b}^{2}}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer5)
  If \[\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0},\] then which relation is correct [RPET 1985; Roorkee 1981; AIEEE 2002]
  A)
                  \[\mathbf{a}=\mathbf{b}=\mathbf{c}=\mathbf{0}\]     done clear
  B)
                  \[\mathbf{a}\,.\,\mathbf{b}=\mathbf{b}\,.\,\mathbf{c}=\mathbf{c}\,.\,\mathbf{a}\] done clear
  C)
                  \[\mathbf{a}\times \mathbf{b}=\mathbf{b}\times \mathbf{c}=\mathbf{c}\times \mathbf{a}\]  done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer6)
  If q be the angle between the vectors a and b and \[|\mathbf{a}\times \mathbf{b}|\,=\mathbf{a}\,.\,\mathbf{b},\] then \[\theta =\]                   [RPET 1990; MP PET 1990; UPSEAT 2003]
  A)
                  \[\pi \]      done clear
  B)
                  \[\frac{\pi }{2}\] done clear
  C)
                  \[\frac{\pi }{4}\] done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer7)
  \[(2\mathbf{a}+3\mathbf{b})\times (5\mathbf{a}+7\mathbf{b})=\]                                           [MP PET 1988]
  A)
                  \[\mathbf{a}\times \mathbf{b}\]          done clear
  B)
                  \[\mathbf{b}\times \mathbf{a}\] done clear
  C)
                  \[\mathbf{a}+\mathbf{b}\]  done clear
  D)
                  \[7\mathbf{a}+10\mathbf{b}\] done clear
  View Solution play_arrow

• question_answer8)
  If a and b are two vectors such that a . b = 0 and \[\mathbf{a}\times \mathbf{b}=\mathbf{0},\] then [IIT Screening 1989; MNR 1988; UPSEAT 2000, 01]
  A)
                  a is parallel to b    done clear
  B)
                  a is perpendicular to b done clear
  C)
                  Either a or b is a null vector done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer9)
  The components of a vector a along and perpendicular to the non-zero vector b are respectively            [IIT 1988]
  A)
                  \[\frac{\mathbf{a}\,.\,\mathbf{b}}{|\mathbf{a}|},\,\frac{|\mathbf{a}\,\times \,\mathbf{b}|}{|\mathbf{a}|}\]        done clear
  B)
                  \[\frac{\mathbf{a}\,.\,\mathbf{b}}{|\mathbf{b}|},\,\frac{|\mathbf{a}\,\times \,\mathbf{b}|}{|\mathbf{b}|}\] done clear
  C)
                  \[\frac{\mathbf{a}\,.\,\mathbf{b}}{|\mathbf{a}|},\,\frac{\mathbf{a}\,.\,\mathbf{b}}{|\mathbf{a}|}\]      done clear
  D)
                  \[\frac{|\mathbf{a}\,\times \,\mathbf{b}|}{|\mathbf{a}|},\,\frac{|\mathbf{a}\,\times \,\mathbf{b}|}{|\mathbf{b}|}\] done clear
  View Solution play_arrow

• question_answer10)
  \[|\,(\mathbf{a}\times \mathbf{b})\,.\,\mathbf{c}\,|\,=\,|\mathbf{a}|\,\,|\mathbf{b}|\,\,|\mathbf{c}|,\] if               [MP PET 1994; BIT Ranchi 1990; IIT 1982; AMU 2002]
  A)
                  \[\mathbf{a}\,.\,\mathbf{b}=\mathbf{b}\,.\,\mathbf{c}=0\]            done clear
  B)
                  \[\mathbf{b}\,.\,\mathbf{c}=\mathbf{c}\,.\,\mathbf{a}=0\] done clear
  C)
                  \[\mathbf{c}\,.\,\mathbf{a}=\mathbf{a}\,.\,\mathbf{b}=0\]            done clear
  D)
                  \[\mathbf{a}\,.\,\mathbf{b}=\mathbf{b}\,.\,\mathbf{c}=\mathbf{c}\,.\,\mathbf{a}=0\] done clear
  View Solution play_arrow

• question_answer11)
  Which of the following is not a property of vectors                                                 [MP PET 1987]
  A)
                  \[\mathbf{u}\times \mathbf{v}=\mathbf{v}\times \mathbf{u}\]     done clear
  B)
                  \[\mathbf{u}\,.\,\mathbf{v}=\mathbf{v}\,.\,\mathbf{u}\] done clear
  C)
                  \[{{(\mathbf{u}\times \mathbf{v})}^{2}}={{\mathbf{u}}^{2}}\,.\,{{\mathbf{v}}^{2}}-{{(\mathbf{u}\,.\,\mathbf{v})}^{2}}\] done clear
  D)
                  \[{{\mathbf{u}}^{2}}=\,|\mathbf{u}{{|}^{2}}\] done clear
  View Solution play_arrow

• question_answer12)
  The number of vectors of unit length perpendicular to vectors \[\mathbf{a}=(1,\,\,1,\,\,0)\] and \[\mathbf{b}=(0,\,\,1,\,\,1)\] is                [BIT Ranchi 1991; IIT 1987; Kurukshetra CEE 1998; DCE 2000; MP PET 2002]
  A)
                  Three done clear
  B)
                  One done clear
  C)
                  Two done clear
  D)
                  Infinite done clear
  View Solution play_arrow

• question_answer13)
  If \[\mathbf{a}=(1,\,\,-1,\,\,1)\] and \[\mathbf{c}=(-1,\,\,-1,\,\,0),\] then the vector b satisfying \[\mathbf{a}\times \mathbf{b}=\mathbf{c}\] and \[\mathbf{a}\,\,.\,\,\mathbf{b}=1\] is           [MP PET 1989]
  A)
                  (1, 0, 0)  done clear
  B)
                  (0, 0, 1) done clear
  C)
                  (0, ?1, 0) done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer14)
  If \[\mathbf{a}\times \mathbf{b}=\mathbf{b}\times \mathbf{c}\ne 0,\] where a, b and c are coplanar vectors, then for some scalar k                    [Roorkee 1985; RPET 1997]
  A)
                  \[\mathbf{a}+\mathbf{c}=k\,\mathbf{b}\] done clear
  B)
                  \[\mathbf{a}+\mathbf{b}=k\,\mathbf{c}\] done clear
  C)
                  \[\mathbf{b}+\mathbf{c}=k\,\mathbf{a}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer15)
  If \[\mathbf{a}\ne \mathbf{0},\,\,\mathbf{b}\ne \mathbf{0},\,\,\mathbf{c}\ne \mathbf{0}\], then true statement is [MP PET 1991]
  A)
                  \[\mathbf{a}\times (\mathbf{b}+\mathbf{c})=(\mathbf{c}+\mathbf{b})\times \mathbf{a}\]    done clear
  B)
                  \[\mathbf{a}\,.\,(\mathbf{b}+\mathbf{c})=-(\mathbf{b}+\mathbf{c})\,.\,\mathbf{a}\] done clear
  C)
                  \[\mathbf{a}\times (\mathbf{b}-\mathbf{c})=(\mathbf{c}-\mathbf{b})\times \mathbf{a}\] done clear
  D)
                  \[\mathbf{a}\,.\,(\mathbf{b}-\mathbf{c})=(\mathbf{c}-\mathbf{b})\,.\,\mathbf{a}\] done clear
  View Solution play_arrow

• question_answer16)
  Let a and b be two non-collinear unit vectors. If \[\mathbf{u}=\mathbf{a}-(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{b}\] and \[\mathbf{v}=\mathbf{a}\times \mathbf{b},\] then | v | is     [IIT 1999]
  A)
                  | u |       done clear
  B)
                  | u |+| u . a | done clear
  C)
                  | u |+| u . b |     done clear
  D)
                  | u |+ u . (a+b) done clear
  View Solution play_arrow

• question_answer17)
  If \[\mathbf{a}\times \mathbf{b}=\mathbf{b}\times \mathbf{c}\ne 0\] and \[\mathbf{a}+\mathbf{c}\ne 0,\] then                 [RPET 1999]
  A)
                  \[(\mathbf{a}+\mathbf{c})\,\bot \,\mathbf{b}\] done clear
  B)
                  \[(\mathbf{a}+\mathbf{c})\,\,|\,\,|\,\,\mathbf{b}\] done clear
  C)
                  \[\mathbf{a}+\mathbf{c}=\mathbf{b}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer18)
  A unit vector perpendicular to the plane determined by the points (1, ? 1, 2), (2, 0, ? 1) and (0, 2, 1) is                                                                 [IIT 1983; MNR 1984]
  A)
                  \[\pm \,\frac{1}{\sqrt{6}}\,(2\mathbf{i}+\mathbf{j}+\mathbf{k})\]              done clear
  B)
                  \[\frac{1}{\sqrt{6}}\,(\mathbf{i}+2\mathbf{j}+\mathbf{k})\] done clear
  C)
                  \[\frac{1}{\sqrt{6}}\,(\mathbf{i}+\mathbf{j}+\mathbf{k})\]          done clear
  D)
                  \[\frac{1}{\sqrt{6}}\,(2\mathbf{i}-\mathbf{j}-\mathbf{k})\] done clear
  View Solution play_arrow

• question_answer19)
  If \[\mathbf{a}=2\mathbf{i}+3\mathbf{j}-5\mathbf{k},\,\,\mathbf{b}=m\mathbf{i}+n\mathbf{j}+12\mathbf{k}\] and \[\mathbf{a}\times \mathbf{b}=0,\] then \[(m,\,\,n)=\]    
  A)
                  \[\left( -\frac{24}{5},\,\frac{36}{5} \right)\]        done clear
  B)
                  \[\left( \frac{24}{5},\,-\frac{36}{5} \right)\] done clear
  C)
                  \[\left( -\frac{24}{5},\,-\frac{36}{5} \right)\]      done clear
  D)
                  \[\left( \frac{24}{5},\,\frac{36}{5} \right)\] done clear
  View Solution play_arrow

• question_answer20)
   A unit vector which is perpendicular to \[\mathbf{i}+2\mathbf{j}-2\mathbf{k}\] and \[-\mathbf{i}+2\mathbf{j}+2\mathbf{k}\] is                                 [MP PET 1992]
  A)
                  \[\frac{1}{\sqrt{5}}\,(2\mathbf{i}-\mathbf{k})\]               done clear
  B)
                  \[\frac{1}{\sqrt{5}}\,(-2\mathbf{i}+\mathbf{k})\] done clear
  C)
                  \[\frac{1}{\sqrt{5}}\,(2\mathbf{i}+\mathbf{j}+\mathbf{k})\]        done clear
  D)
                  \[\frac{1}{\sqrt{5}}\,(2\mathbf{i}+\mathbf{k})\] done clear
  View Solution play_arrow

• question_answer21)
  If \[A\,(-1,\,\,2,\,\,3),\,\,B\,(1,\,\,1,\,\,1)\] and \[C\,(2,\,\,-1,\,\,3)\] are points on a plane. A unit normal vector to the plane ABC is                                                 [BIT Ranchi 1988]
  A)
                  \[\pm \,\left( \frac{2\mathbf{i}+2\mathbf{j}+\mathbf{k}}{3} \right)\]           done clear
  B)
                  \[\pm \,\left( \frac{2\mathbf{i}-2\mathbf{j}+\mathbf{k}}{3} \right)\] done clear
  C)
                  \[\pm \,\left( \frac{2\mathbf{i}-2\mathbf{j}-\mathbf{k}}{3} \right)\]               done clear
  D)
                  \[-\,\left( \frac{2\mathbf{i}+2\mathbf{j}+\mathbf{k}}{3} \right)\] done clear
  View Solution play_arrow

• question_answer22)
  The unit vector perpendicular to the vectors \[6\mathbf{i}+2\mathbf{j}+3\mathbf{k}\] and \[3\mathbf{i}-6\mathbf{j}-2\mathbf{k},\] is                                                 [IIT 1989; RPET 1996]
  A)
                  \[\frac{2\mathbf{i}-3\mathbf{j}+6\mathbf{k}}{7}\]       done clear
  B)
                  \[\frac{2\mathbf{i}-3\mathbf{j}-6\mathbf{k}}{7}\] done clear
  C)
                  \[\frac{2\mathbf{i}+3\mathbf{j}-6\mathbf{k}}{7}\] done clear
  D)
                  \[\frac{2\mathbf{i}+3\mathbf{j}+6\mathbf{k}}{7}\] done clear
  View Solution play_arrow

• question_answer23)
  For any two vectors a and b, \[{{(\mathbf{a}\times \mathbf{b})}^{2}}\] is equal to [Roorkee 1975, 79, 81, 85]
  A)
                  \[{{a}^{2}}-{{b}^{2}}\] done clear
  B)
                  \[{{a}^{2}}+{{b}^{2}}\] done clear
  C)
                  \[{{a}^{2}}{{b}^{2}}-{{(\mathbf{a}\,.\,\mathbf{b})}^{2}}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer24)
  The unit vector perpendicular to \[3\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and \[12\mathbf{i}+5\mathbf{j}-5\mathbf{k},\] is [Roorkee 1979; RPET 1989, 91]
  A)
                  \[\frac{5\mathbf{i}-3\mathbf{j}+9\mathbf{k}}{\sqrt{115}}\]       done clear
  B)
                  \[\frac{5\mathbf{i}+3\mathbf{j}-9\mathbf{k}}{\sqrt{115}}\] done clear
  C)
                  \[\frac{-5\mathbf{i}+3\mathbf{j}-9\mathbf{k}}{\sqrt{115}}\]      done clear
  D)
                  \[\frac{5\mathbf{i}+3\mathbf{j}+9\mathbf{k}}{\sqrt{115}}\] done clear
  View Solution play_arrow

• question_answer25)
  The sine of the angle between the two vectors \[3\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and \[12\mathbf{i}+5\mathbf{j}-5\mathbf{k}\] will be            [Roorkee 1978]
  A)
                  \[\frac{\sqrt{115}}{\sqrt{14}\sqrt{194}}\]        done clear
  B)
                  \[\frac{51}{\sqrt{14}\sqrt{144}}\] done clear
  C)
                  \[\frac{\sqrt{64}}{\sqrt{14}\sqrt{194}}\]          done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer26)
  For any two vectors a and b, if \[\mathbf{a}\times \mathbf{b}=\mathbf{0},\] then [Roorkee 1984]
  A)
                  \[\mathbf{a}=\mathbf{0}\] done clear
  B)
                  \[\mathbf{b}=\mathbf{0}\] done clear
  C)
                  Not parallel done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer27)
  If a and b are two vectors, then \[{{(\mathbf{a}\times \mathbf{b})}^{2}}\] equals [Roorkee 1975, 79, 81, 85]
  A)
                  \[\left| \,\begin{matrix}    \mathbf{a}\,\,.\,\,\mathbf{b} & \mathbf{a}\,\,.\,\,\mathbf{a}  \\    \mathbf{b}\,\,.\,\,\mathbf{b} & \mathbf{b}\,\,.\,\,\mathbf{a}  \\ \end{matrix}\, \right|\]          done clear
  B)
                  \[\left| \,\begin{matrix}    \mathbf{a}\,\,.\,\,\mathbf{a} & \mathbf{a}\,\,.\,\,\mathbf{b}  \\    \mathbf{b}\,\,.\,\,\mathbf{a} & \mathbf{b}\,\,.\,\,\mathbf{b}  \\ \end{matrix}\, \right|\] done clear
  C)
                  \[\left| \,\begin{matrix}    \mathbf{a}\,\,.\,\,\mathbf{b}  \\    \mathbf{b}\,\,.\,\,\mathbf{a}  \\ \end{matrix}\, \right|\]          done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer28)
  For any vectors a, b, c  \[\mathbf{a}\times (\mathbf{b}+\mathbf{c})+\mathbf{b}\times (\mathbf{c}+\mathbf{a})+\mathbf{c}\times (\mathbf{a}+\mathbf{b})=\] [Roorkee 1981; Kerala (Engg.) 2002]
  A)
                  0 done clear
  B)
                  \[\mathbf{a}+\mathbf{b}+\mathbf{c}\] done clear
  C)
                  \[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\] done clear
  D)
                  \[\mathbf{a}\times \mathbf{b}\times \mathbf{c}\] done clear
  View Solution play_arrow

• question_answer29)
  If \[\mathbf{a}\,.\,\mathbf{b}=\mathbf{a}\,.\,\mathbf{c},\,\,\mathbf{a}\,\times \mathbf{b}=\mathbf{a}\times \mathbf{c}\] and \[\mathbf{a}\ne \mathbf{0},\] then     [RPET 1990]
  A)
                  \[\mathbf{b}=\mathbf{0}\] done clear
  B)
                  \[\mathbf{b}\ne \mathbf{c}\] done clear
  C)
                  \[\mathbf{b}=\mathbf{c}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer30)
  If \[|\mathbf{a}|\,=2,\,\,|\mathbf{b}|\,=5\] and \[|\mathbf{a}\times \mathbf{b}|\,=8,\] then a . b is equal to                 [AI CBSE 1984; RPET 1991]
  A)
                  0 done clear
  B)
                  2 done clear
  C)
                  4 done clear
  D)
                  6 done clear
  View Solution play_arrow

• question_answer31)
  If \[|\mathbf{a}\,.\,\mathbf{b}|\,=3\] and \[|\mathbf{a}\times \mathbf{b}|\,=4,\] then the angle between a and b is
  A)
                  \[{{\cos }^{-1}}\frac{3}{4}\]            done clear
  B)
                  \[{{\cos }^{-1}}\frac{3}{5}\] done clear
  C)
                  \[{{\cos }^{-1}}\frac{4}{5}\]            done clear
  D)
                  \[\frac{\pi }{4}\] done clear
  View Solution play_arrow

• question_answer32)
  If \[\mathbf{a}=2\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and \[\mathbf{b}=6\mathbf{i}-3\mathbf{j}+2\mathbf{k},\] then the value of \[\mathbf{a}\times \mathbf{b}\] is                      [MNR 1978; RPET 2001]
  A)
                  \[2\mathbf{i}+2\mathbf{j}-\mathbf{k}\]           done clear
  B)
                  \[6\mathbf{i}-3\mathbf{j}+2\mathbf{k}\] done clear
  C)
                  \[\mathbf{i}-10\mathbf{j}-18\mathbf{k}\]        done clear
  D)
                  \[\mathbf{i}+\mathbf{j}+\mathbf{k}\] done clear
  View Solution play_arrow

• question_answer33)
  The scalars l and m such that \[l\mathbf{a}+m\mathbf{b}=\mathbf{c},\] where a, b and c are given vectors, are equal to
  A)
                  \[l=\frac{(\mathbf{c}\times \mathbf{b})\,.\,(\mathbf{a}\times \mathbf{b})}{{{(\mathbf{a}\times \mathbf{b})}^{2}}},\,\,m=\frac{(\mathbf{c}\times \mathbf{a})\,.\,(\mathbf{b}\times \mathbf{a})}{{{(\mathbf{b}\times \mathbf{a})}^{2}}}\] done clear
  B)
                  \[l=\frac{(\mathbf{c}\times \mathbf{b})\,.\,(\mathbf{a}\times \mathbf{b})}{(\mathbf{a}\times \mathbf{b})},\,\,m=\frac{(\mathbf{c}\times \mathbf{a})\,.\,(\mathbf{b}\times \mathbf{a})}{(\mathbf{b}\times \mathbf{a})}\] done clear
  C)
                  \[l=\frac{(\mathbf{c}\times \mathbf{b})\,\times \,(\mathbf{a}\times \mathbf{b})}{{{(\mathbf{a}\times \mathbf{b})}^{2}}},\,\,m=\frac{(\mathbf{c}\times \mathbf{a})\,\times \,(\mathbf{b}\times \mathbf{a})}{(\mathbf{b}\times \mathbf{a})}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer34)
  \[|\mathbf{a}\times \mathbf{i}{{|}^{2}}+|\mathbf{a}\times \mathbf{j}{{|}^{2}}+|\mathbf{a}\times \mathbf{k}{{|}^{2}}=\]                 [EAMCET 1988; MP PET 1994, 2004; RPET 2000;   Pb. CET 2001; Orissa JEE 2003; AIEEE 2005]
  A)
                  \[|\mathbf{a}{{|}^{2}}\]  done clear
  B)
                  \[2\,\,|\mathbf{a}{{|}^{2}}\] done clear
  C)
                  \[3\,\,|\mathbf{a}{{|}^{2}}\]            done clear
  D)
                  \[4\,\,|\mathbf{a}{{|}^{2}}\] done clear
  View Solution play_arrow

• question_answer35)
  A unit vector perpendicular to the plane determined by the points \[P\,(1,\,\,-1,\,\,2),\,\,Q\,(2,\,\,0,\,-1)\] and \[R\,(0,\,\,2,\,\,1)\] is           [IIT 1994]
  A)
                  \[\frac{2\mathbf{i}-\mathbf{j}+\mathbf{k}}{\sqrt{6}}\] done clear
  B)
                  \[\frac{2\mathbf{i}+\mathbf{j}+\mathbf{k}}{\sqrt{6}}\] done clear
  C)
                  \[\frac{-2\mathbf{i}+\mathbf{j}+\mathbf{k}}{\sqrt{6}}\]             done clear
  D)
                  \[\frac{2\mathbf{i}+\mathbf{j}-\mathbf{k}}{\sqrt{6}}\] done clear
  View Solution play_arrow

• question_answer36)
  A unit vector perpendicular to the vector \[4\mathbf{i}-\mathbf{j}+3\mathbf{k}\] and \[-2\mathbf{i}+\mathbf{j}-2\mathbf{k}\] is                                  [MNR 1995]
  A)
                  \[\frac{1}{3}\,(\mathbf{i}-2\mathbf{j}+2\mathbf{k})\]   done clear
  B)
                  \[\frac{1}{3}\,(-\mathbf{i}+2\mathbf{j}+2\mathbf{k})\] done clear
  C)
                  \[\frac{1}{3}\,(2\mathbf{i}+\mathbf{j}+2\mathbf{k})\] done clear
  D)
                  \[\frac{1}{3}\,(2\mathbf{i}-2\mathbf{j}+2\mathbf{k})\] done clear
  View Solution play_arrow

• question_answer37)
  Given \[\mathbf{a}=\mathbf{i}+\mathbf{j}-\mathbf{k},\,\,\mathbf{b}=-\mathbf{i}+2\mathbf{j}+\mathbf{k}\] and \[\mathbf{c}=-\mathbf{i}+2\mathbf{j}-\mathbf{k}.\] A unit vector perpendicular to both \[\mathbf{a}+\mathbf{b}\] and \[\mathbf{b}+\mathbf{c}\] is                                                 [Karnataka CET 1993]
  A)
                  i done clear
  B)
                  j done clear
  C)
                  k done clear
  D)
                  \[\frac{\mathbf{i}+\mathbf{j}+\mathbf{k}}{\sqrt{3}}\] done clear
  View Solution play_arrow

• question_answer38)
  The vectors \[\mathbf{c},\,\,\,\mathbf{a}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\] and \[\mathbf{b}=\mathbf{j}\] are such that       a, c, b form a right handed system, then c is                 [DCE 1999]
  A)
                  \[z\mathbf{i}-x\mathbf{k}\] done clear
  B)
                  0 done clear
  C)
                  \[y\mathbf{j}\] done clear
  D)
                  \[-z\mathbf{i}+x\mathbf{k}\] done clear
  View Solution play_arrow

• question_answer39)
  If A, B, C, D are any four points in space, then \[|\overrightarrow{AB}\times \overrightarrow{CD}+\overrightarrow{BC}\times \overrightarrow{AD}+\overrightarrow{CA}\times \overrightarrow{BD}|\] is equal to
  A)
                  \[2\Delta \] done clear
  B)
                  \[4\Delta \] done clear
  C)
                  \[3\Delta \] done clear
  D)
                  \[5\Delta \] (where D denotes the area of \[\Delta ABC\]) done clear
  View Solution play_arrow

• question_answer40)
  If \[{{(\mathbf{a}\times \mathbf{b})}^{2}}+{{(\mathbf{a}\,\,.\,\,\mathbf{b})}^{2}}=144\] and \[|\mathbf{a}|\,=4,\] then \[|\mathbf{b}|\,=\]                                                 [EAMCET 1994]
  A)
                  16 done clear
  B)
                  8 done clear
  C)
                  3 done clear
  D)
                  12 done clear
  View Solution play_arrow

• question_answer41)
  \[\mathbf{r}\times \mathbf{a}=\mathbf{b}\times \mathbf{a};\,\,\mathbf{r}\times \mathbf{b}=\mathbf{a}\times \mathbf{b};\,\,\mathbf{a}\ne 0;\,\,\mathbf{b}\ne 0;\,\,\mathbf{a}\ne \lambda \mathbf{b},\,\,\]a  is not perpendicular to b, then \[\mathbf{r}=\]                                     [EAMCET 1993]
  A)
                  \[\mathbf{a}-\mathbf{b}\] done clear
  B)
                  \[\mathbf{a}+\mathbf{b}\] done clear
  C)
                  \[\mathbf{a}\times \mathbf{b}+\mathbf{a}\] done clear
  D)
                  \[\mathbf{a}\times \mathbf{b}+\mathbf{b}\] done clear
  View Solution play_arrow

• question_answer42)
  If \[\mathbf{i},\,\,\mathbf{j},\,\,\mathbf{k}\] are unit orthonormal vectors and a is a vector, if \[\mathbf{a}\times \mathbf{r}=\mathbf{j},\] then a . r is                                           [EAMCET 1990]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  ? 1 done clear
  D)
                  Arbitrary scalar done clear
  View Solution play_arrow

• question_answer43)
  A unit vector perpendicular to each of the vector \[2\mathbf{i}-\mathbf{j}+\mathbf{k}\] and \[3\mathbf{i}+4\mathbf{j}-\mathbf{k}\] is equal to                                  [MP PET 2003]
  A)
                  \[\frac{(-3\mathbf{i}+5\mathbf{j}+11\mathbf{k})}{\sqrt{155}}\]               done clear
  B)
                  \[\frac{(3\mathbf{i}-5\mathbf{j}+11\mathbf{k})}{\sqrt{155}}\] done clear
  C)
                  \[\frac{(6\mathbf{i}-4\mathbf{j}-\mathbf{k})}{\sqrt{53}}\]           done clear
  D)
                  \[\frac{(5\mathbf{i}+3\mathbf{j})}{\sqrt{34}}\] done clear
  View Solution play_arrow

• question_answer44)
  If \[\overrightarrow{A}=3\mathbf{i}+\mathbf{j}+2\mathbf{k}\] and \[\overrightarrow{B}=2\mathbf{i}-2\mathbf{j}+4\mathbf{k}\] and q is the angle between \[\overrightarrow{A}\] and \[\overrightarrow{B},\] then the value of \[\sin \theta \] is
  A)
                  \[\frac{2}{\sqrt{7}}\] done clear
  B)
                  \[\sqrt{\frac{2}{7}}\] done clear
  C)
                  \[\frac{4}{\sqrt{7}}\] done clear
  D)
                  \[\frac{3}{\sqrt{7}}\] done clear
  View Solution play_arrow

• question_answer45)
  A unit vector perpendicular to vector c and coplanar with vectors a and b is  [MP PET 1999]
  A)
                  \[\frac{\mathbf{a}\times (\mathbf{b}\times \mathbf{c})}{|\mathbf{a}\times (\mathbf{b}\times \mathbf{c})|}\] done clear
  B)
                  \[\frac{\mathbf{b}\times (\mathbf{c}\times \mathbf{a})}{|\mathbf{b}\times (\mathbf{c}\times \mathbf{a})|}\] done clear
  C)
                  \[\frac{\mathbf{c}\times (\mathbf{a}\times \mathbf{b})}{|\mathbf{c}\times (\mathbf{a}\times \mathbf{b})|}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer46)
  \[|\mathbf{a}\times \mathbf{b}{{|}^{2}}+\,{{(\mathbf{a}\,.\,\mathbf{b})}^{2}}=\]                 [MP PET 1989, 97, 2004]
  A)
                  \[(\mathbf{a}\times \mathbf{a})\,.\,(\mathbf{b}\times \mathbf{b})\] done clear
  B)
                  \[(\mathbf{a}\,.\,\mathbf{a})\,(\mathbf{b}\,.\,\mathbf{b})\] done clear
  C)
                  \[|\,(\mathbf{a}\times \mathbf{b})\,|\,\,(\mathbf{a}\,.\,\mathbf{b})\] done clear
  D)
                  \[2\,(\mathbf{a}\,.\,\mathbf{b})\,(\mathbf{a}\,.\,\mathbf{b})\] done clear
  View Solution play_arrow

• question_answer47)
  If the position vectors of three points \[A,B\] and \[C\] are respectively i + j + k, 2i + 3j ? 4k and 7i + 4j + 9k, then the unit vector to the plane containing the triangle \[ABC\] is                                                 [DCE 1999]
  A)
                  31i ? 18j ? 9k done clear
  B)
                  \[\frac{31i-38j-9 k}{\sqrt{2486}}\] done clear
  C)
                  \[\frac{31i+18j+9k}{\sqrt{2486}}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer48)
  If a, b, c are position vector of vertices of a triangle \[ABC\], then unit vector perpendicular to its plane is              [RPET 1999]
  A)
                  \[a\times b+b\times c+c\times a\] done clear
  B)
                  \[\frac{a\times b+b\times c+c\times a}{|a\times b+b\times c+c\times a|}\] done clear
  C)
                  \[\frac{a\times b}{|a\times b|}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer49)
  If \[\theta \] is the angle between the vectors a and b, then \[\frac{|a\times b|}{|a\,.\,b|}\] equal to                                          [Karnataka CET 1999]
  A)
                  \[\tan \theta \] done clear
  B)
                  \[-\tan \theta \] done clear
  C)
                  \[\cot \theta \] done clear
  D)
                  \[-\cot \theta \] done clear
  View Solution play_arrow

• question_answer50)
  If the vectors \[\mathbf{a},\,\mathbf{b}\] and c are represented by the sides \[BC,\,CA\] and \[AB\] respectively of the \[\Delta ABC\], then [IIT Screening 2000]
  A)
                  \[\mathbf{a}\,\mathbf{.}\,\mathbf{b}+\mathbf{b}\,\mathbf{.}\,\mathbf{c}+\mathbf{c}\,\mathbf{.}\,\mathbf{a}=0\] done clear
  B)
                  \[\mathbf{a}\times \mathbf{b}=\mathbf{b}\times \mathbf{c}=\mathbf{c}\times \mathbf{a}\] done clear
  C)
                  \[\mathbf{a}\,\mathbf{.}\,\mathbf{b}=\mathbf{b}\,\mathbf{.}\,\mathbf{c}=\mathbf{c}\,\mathbf{.}\,\mathbf{a}\] done clear
  D)
                  \[\mathbf{a}\times \mathbf{b}=\mathbf{b}\times \mathbf{c}=\mathbf{c}\times \mathbf{a}=0\] done clear
  View Solution play_arrow

• question_answer51)
  A vector perpendicular to both of the vectors \[i+j+k\] and \[i+j\] is                                             [RPET 2000]
  A)
                  i + j done clear
  B)
                  i ? j done clear
  C)
                  \[c(i-j)\], c is a scalar done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer52)
  A unit vector perpendicular to the plane of \[a=2i-6j-3k\], \[b=4i+3j-k\] is  [MP PET 2000]
  A)
                  \[\frac{4i+3j-k}{\sqrt{26}}\] done clear
  B)
                  \[\frac{2i-6j-3k}{7}\] done clear
  C)
                  \[\frac{3i-2j+6k}{7}\] done clear
  D)
                  \[\frac{2i-3j-6k}{7}\] done clear
  View Solution play_arrow

• question_answer53)
  The unit vector perpendicular to both the vectors i ? 2j + 3k  and i + 2j ? k is [DCE 2001]
  A)
                  \[\frac{1}{\sqrt{3}}(-i+j+k)\] done clear
  B)
                  \[(-\mathbf{i}+\mathbf{j}+\mathbf{k})\] done clear
  C)
                  \[\frac{(i+j-k)}{\sqrt{3}}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer54)
  The unit vector perpendicular to the vectors \[\mathbf{i}-\mathbf{j}+\mathbf{k}\] and \[2\mathbf{i}+3\mathbf{j}-\mathbf{k}\] is [Karnataka CET 2001]
  A)
                  \[\frac{-2i+3j+5k}{\sqrt{30}}\] done clear
  B)
                  \[\frac{-2\mathbf{i}+5\mathbf{j}+6\mathbf{k}}{\sqrt{38}}\]\[\] done clear
  C)
                  \[\frac{-2i+3j+5k}{\sqrt{38}}\] done clear
  D)
                  \[\frac{-2i+4j+5k}{\sqrt{38}}\] done clear
  View Solution play_arrow

• question_answer55)
  If \[\mathbf{a}=2\mathbf{i}-3\,\mathbf{j}-\mathbf{k}\] and \[\mathbf{b}=\mathbf{i}+4\mathbf{j}-2\mathbf{k}\], then \[\mathbf{a}\times \mathbf{b}\] is [MP PET 2001]
  A)
                  10i + 2j + 11k done clear
  B)
                  10i + 3j + 11k done clear
  C)
                  10i ? 3j + 11k done clear
  D)
                  10i ? 3j ? 10k done clear
  View Solution play_arrow

• question_answer56)
  If \[|\mathbf{a}|\,=4,\,|\mathbf{b}|\,=2\]  and the angle between a and b  is \[\frac{\pi }{6}\], then \[{{(\mathbf{a}\times \mathbf{b})}^{2}}\] is equal to [AIEEE 2002]       
  A)
                  48 done clear
  B)
                  16 done clear
  C)
                  8 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer57)
  If \[\mathbf{a}=2\mathbf{i}+4\mathbf{j}-5\mathbf{k}\] and \[\mathbf{b}=\mathbf{i}+2\mathbf{j}+3\mathbf{k}\], then \[|\mathbf{a}\times \mathbf{b}|\]  is                                                 [UPSEAT 2002]
  A)
                  \[11\sqrt{5}\] done clear
  B)
                  \[11\sqrt{3}\] done clear
  C)
                  \[11\sqrt{7}\] done clear
  D)
                  \[11\sqrt{2}\]         done clear
  View Solution play_arrow

• question_answer58)
  The unit vector perpendicular to both \[\mathbf{i}+\mathbf{j}\] and \[\mathbf{j}+\mathbf{k}\] is                                 [Kerala (Engg.) 2002]
  A)
                  i ? j + k done clear
  B)
                  i + j + k done clear
  C)
                  \[\frac{\mathbf{i}+\mathbf{j}-\mathbf{k}}{\sqrt{3}}\]   done clear
  D)
                  \[\frac{\mathbf{i}-\mathbf{j}+\mathbf{k}}{\sqrt{3}}\] done clear
  View Solution play_arrow

• question_answer59)
  A unit vector in the plane of the vectors \[2\mathbf{i}+\mathbf{j}+\mathbf{k},\,\] \[\,\mathbf{i}-\mathbf{j}+\mathbf{k}\] and orthogonal to \[5\mathbf{i}+2\mathbf{j}+6\mathbf{k}\] is  [IIT Screening 2004]
  A)
                  \[\frac{6\mathbf{i}-5\mathbf{k}}{\sqrt{61}}\] done clear
  B)
                  \[\frac{3\mathbf{j}-\mathbf{k}}{\sqrt{10}}\] done clear
  C)
                  \[\frac{2\mathbf{i}-5\mathbf{j}}{\sqrt{29}}\] done clear
  D)
                  \[\frac{2\mathbf{i}+\mathbf{j}-2\mathbf{k}}{3}\] done clear
  View Solution play_arrow

• question_answer60)
  Let \[\mathbf{a},\,\mathbf{b},\,\mathbf{c}\] be three vectors such that \[\mathbf{a}\ne 0,\] and \[\mathbf{a}\times \mathbf{b}=2\mathbf{a}\times \mathbf{c},\,\,|\mathbf{a}|\,=\,|\mathbf{c}|\,=\,1,\,|\mathbf{b}|\,=4\] and \[|\mathbf{b}\times \mathbf{c}|\,=15.\] If \[\mathbf{b}-2\mathbf{c}=\lambda \mathbf{a},\] then l equals to [Orissa JEE 2004]
  A)
                  1 done clear
  B)
                  \[\pm \,4\] done clear
  C)
                  3 done clear
  D)
                  ? 2 done clear
  View Solution play_arrow

• question_answer61)
  The area of a triangle whose vertices are \[A\,(1,\,-1,\,2),\] \[B\,(2,\,1,\,-1)\] and \[C\,(3,\,-1,\,2)\] is [MNR 1983; IIT 1983]
  A)
                  13 done clear
  B)
                  \[\sqrt{13}\] done clear
  C)
                  6 done clear
  D)
                  \[\sqrt{6}\] done clear
  View Solution play_arrow

• question_answer62)
  If vertices of a triangle are \[A(1,\,-1,\,2),\,B(2,\,0,\,-1)\] and \[C(0,\,2,\,1),\] then the area of a triangle is    [RPET 2000]
  A)
                  \[\sqrt{6}\] done clear
  B)
                  \[2\sqrt{6}\] done clear
  C)
                  \[3\sqrt{6}\] done clear
  D)
                  \[4\sqrt{6}\] done clear
  View Solution play_arrow

• question_answer63)
  The area of triangle whose vertices are \[(1,\,2,\,3),\,(2,\,5,\,-1)\] and \[(-1,\,1,\,2)\] is [Kerala (Engg.)  2002]
  A)
                  150 sq. unit done clear
  B)
                  145 sq. unit done clear
  C)
                  \[\frac{\sqrt{155}}{2}\] sq. unit done clear
  D)
                  \[\frac{155}{2}\] sq. unit done clear
  View Solution play_arrow

• question_answer64)
  The area of a parallelogram whose two adjacent sides are represented by the vector \[3\mathbf{i}-\mathbf{k}\] and \[\mathbf{i}+2\mathbf{j}\] is   [MNR 1981]
  A)
                  \[\frac{1}{2}\sqrt{17}\] done clear
  B)
                  \[\frac{1}{2}\sqrt{14}\] done clear
  C)
                  \[\sqrt{41}\] done clear
  D)
                  \[\frac{1}{2}\sqrt{7}\] done clear
  View Solution play_arrow

• question_answer65)
  The area of the parallelogram whose diagonals are \[\mathbf{a}=3\,\mathbf{i}+\mathbf{j}-2\mathbf{k}\] and \[\mathbf{b}=\mathbf{i}-3\,\mathbf{j}+4\,\mathbf{k}\] is [MP PET 1988, 93; MNR 1985]
  A)
                  \[10\sqrt{3}\] done clear
  B)
                  \[5\sqrt{3}\] done clear
  C)
                  8 done clear
  D)
                  4 done clear
  View Solution play_arrow

• question_answer66)
  The position vectors of the points A, B and C are \[\mathbf{i}+\mathbf{j},\,\,\mathbf{j}+\mathbf{k}\] and \[\mathbf{k}+\mathbf{i}\] respectively. The vector area of the \[\Delta ABC=\pm \,\frac{1}{2}\overrightarrow{\alpha }\] where \[\overrightarrow{\alpha }=\] [MP PET 1989]
  A)
                  \[-\mathbf{i}+\mathbf{j}+\mathbf{k}\] done clear
  B)
                  \[\mathbf{i}-\mathbf{j}+\mathbf{k}\] done clear
  C)
                  \[\mathbf{i}+\mathbf{j}-\mathbf{k}\] done clear
  D)
                  \[\mathbf{i}+\mathbf{j}+\mathbf{k}\] done clear
  View Solution play_arrow

• question_answer67)
  If \[\overrightarrow{OA}=3\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and \[\overrightarrow{OB}=\mathbf{i}+3\mathbf{j}+\mathbf{k}\], then the area of the triangle OAB is   
  A)
                  \[\sqrt{15}\] done clear
  B)
                  \[3\sqrt{5}\] done clear
  C)
                  \[\frac{3}{2}\sqrt{10}\] done clear
  D)
                  \[\frac{5\sqrt{5}}{3}\] done clear
  View Solution play_arrow

• question_answer68)
  Let a, b, c be the position vectors of the vertices of a triangle ABC. The vector area of triangle ABC is   [MP PET 1990; EAMCET 2003]
  A)
                  \[\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a}\] done clear
  B)
                  \[\frac{1}{4}(\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a})\] done clear
  C)
                  \[\frac{1}{2}(\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a})\] done clear
  D)
                  \[\mathbf{b}\times \mathbf{a}+\mathbf{c}\times \mathbf{b}+\mathbf{a}\times \mathbf{c}\] done clear
  View Solution play_arrow

• question_answer69)
  If \[|\mathbf{a}|\,=2,\,\,|\mathbf{b}|\,=3\] and a, b are mutually perpendicular, then the area of the triangle whose vertices are \[\mathbf{0},\,\,\mathbf{a}+\mathbf{b},\,\,\mathbf{a}-\mathbf{b}\] is
  A)
                  5 done clear
  B)
                  1 done clear
  C)
                  6 done clear
  D)
                  8 done clear
  View Solution play_arrow

• question_answer70)
  If \[\mathbf{i}+2\mathbf{j}+3\mathbf{k}\] and \[3\mathbf{i}-2\mathbf{j}+\mathbf{k}\] represents the adjacent sides of a parallelogram, then the area of this parallelogram is [Roorkee 1978, 79; MP PET 1990; RPET 1988, 89, 91]
  A)
                  \[4\sqrt{3}\] done clear
  B)
                  \[6\sqrt{3}\] done clear
  C)
                  \[8\sqrt{3}\] done clear
  D)
                  \[16\sqrt{3}\] done clear
  View Solution play_arrow

• question_answer71)
  If \[3\mathbf{i}+4\mathbf{j}\] and \[-5\mathbf{i}+7\mathbf{j}\] are the vector sides of any triangle, then its area is given by [RPET 1987, 90]
  A)
                  41 done clear
  B)
                  47 done clear
  C)
                  \[\frac{41}{2}\] done clear
  D)
                  \[\frac{47}{2}\] done clear
  View Solution play_arrow

• question_answer72)
  If the vectors  \[\mathbf{i}-3\mathbf{j}+2\mathbf{k}\], \[-\mathbf{i}+2\mathbf{j}\] represents the diagonals of a parallelogram, then its area will be              [Roorkee 1976]
  A)
                  \[\sqrt{21}\] done clear
  B)
                  \[\frac{\sqrt{21}}{2}\] done clear
  C)
                  \[2\sqrt{21}\] done clear
  D)
                  \[\frac{\sqrt{21}}{4}\] done clear
  View Solution play_arrow

• question_answer73)
  The area of the parallelogram whose diagonals are the vectors \[2\mathbf{a}-\mathbf{b}\] and \[4\mathbf{a}-5\mathbf{b},\] where a and b are the unit vectors forming an angle of \[{{45}^{o}},\] is
  A)
                  \[3\sqrt{2}\] done clear
  B)
                  \[\frac{3}{\sqrt{2}}\] done clear
  C)
                  \[\sqrt{2}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer74)
  The area of a parallelogram whose adjacent sides are \[\mathbf{i}-2\mathbf{j}+3\mathbf{k}\] and \[2\mathbf{i}+\mathbf{j}-4\mathbf{k},\] is [MP PET 1996, 2000]
  A)
                  \[5\sqrt{3}\] done clear
  B)
                  \[10\sqrt{3}\] done clear
  C)
                  \[5\sqrt{6}\] done clear
  D)
                  \[10\sqrt{6}\] done clear
  View Solution play_arrow

• question_answer75)
  If the diagonals of a parallelogram are represented by the vectors \[3\mathbf{i}+\mathbf{j}-2\mathbf{k}\] and \[\mathbf{i}+3\mathbf{j}-4\mathbf{k},\] then its area in square unit is [MP PET 1998]
  A)
                  \[5\sqrt{3}\] done clear
  B)
                  \[6\sqrt{3}\] done clear
  C)
                  \[\sqrt{26}\] done clear
  D)
                  \[\sqrt{42}\] done clear
  View Solution play_arrow

• question_answer76)
  The area of a parallelogram whose adjacent sides are given by the vectors \[\mathbf{i}+2\mathbf{j}+3\mathbf{k}\] and \[-3\mathbf{i}-2\mathbf{j}+\mathbf{k}\] (in square unit) is [Karnataka CET 2001; Pb. CET 2004]
  A)
                  \[\sqrt{180}\] done clear
  B)
                  \[\sqrt{140}\] done clear
  C)
                  \[\sqrt{80}\] done clear
  D)
                  \[\sqrt{40}\] done clear
  View Solution play_arrow

• question_answer77)
  If \[\mathbf{a}=\mathbf{i}+\mathbf{j}+\mathbf{k},\,\mathbf{b}=\mathbf{i}+3\mathbf{j}+5\mathbf{k}\] and \[\mathbf{c}=7\mathbf{i}+9\mathbf{j}+11\mathbf{k}\], then the area of the parallelogram having diagonals a + b and    b + c is [Kurukshetra CEE 2002]
  A)
                  \[4\sqrt{6}\] done clear
  B)
                  \[\frac{1}{2}\sqrt{21}\] done clear
  C)
                  \[\frac{\sqrt{6}}{2}\] done clear
  D)
                  \[\sqrt{6}\] done clear
  View Solution play_arrow

• question_answer78)
  The area of the parallelogram whose diagonals are  \[\frac{3}{2}\mathbf{i}+\frac{1}{2}\mathbf{j}-\mathbf{k}\] and \[2\mathbf{i}-6\mathbf{j}+8\mathbf{k}\]  is                              [UPSEAT 2002]
  A)
                  \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=bc+ca+ab\] done clear
  B)
                  \[5\sqrt{2}\] done clear
  C)
                  \[({{a}^{2}}+ab,\,{{b}^{2}}+ab,\,-ab)\] done clear
  D)
                  \[(-bc,\,{{b}^{2}}+bc,\,{{c}^{2}}+bc),\] done clear
  View Solution play_arrow

• question_answer79)
  The area of the triangle having vertices as  \[\mathbf{i}-2\mathbf{j}+3\mathbf{k},\] \[\,-2\mathbf{i}+3\mathbf{j}+\mathbf{k}\] , \[4\mathbf{i}-7\mathbf{j}+7\mathbf{k}\] is                     [MP PET 2004]
  A)
                  26 done clear
  B)
                  11 done clear
  C)
                  36 done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer80)
  The area of the parallelogram whose adjacent sides are \[\mathbf{i}-\mathbf{k}\] and \[2\mathbf{j}+3\mathbf{k}\] is                                       [UPSEAT 2004]
  A)
                  2 done clear
  B)
                  4 done clear
  C)
                  \[\sqrt{17}\] done clear
  D)
                  \[2\sqrt{13}\] done clear
  View Solution play_arrow

• question_answer81)
  The moment of the force \[\overrightarrow{F}\] acting at a point P, about the point C is                                            [MP PET 1987]
  A)
                  \[\overrightarrow{F}\times \overrightarrow{CP}\] done clear
  B)
                  \[\overrightarrow{CP}\,.\,\overrightarrow{F}\] done clear
  C)
                  A vector having the same direction as \[\overrightarrow{F}\] done clear
  D)
              \[\overrightarrow{CP}\times \overrightarrow{F}\] done clear
  View Solution play_arrow

• question_answer82)
  Three forces \[\mathbf{i}+2\,\mathbf{j}-3\,\mathbf{k},\,\,2\,\mathbf{i}+3\,\mathbf{j}+4\,\mathbf{k}\] and \[\mathbf{i}-\mathbf{j}+\mathbf{k}\] are acting on a particle at the point (0, 1, 2). The magnitude of the moment of the forces about the point \[(1,\,-2,\,0)\] is                 [MNR 1983]
  A)
                  \[2\sqrt{35}\] done clear
  B)
                  \[6\sqrt{10}\] done clear
  C)
                  \[4\sqrt{17}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer83)
  Let the points A, B and P be (? 2, 2, 4), (2, 6, 3) and (1,2,1) respectively. The magnitude of the moment of the force represented by \[\overrightarrow{AB}\] and acting at A about P is [MP PET 1987]
  A)
                  15 done clear
  B)
                  \[3\sqrt{41}\] done clear
  C)
                  \[3\sqrt{57}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer84)
  The moment of a force represented by \[\overrightarrow{F}=\mathbf{i}+2\mathbf{j}+3\mathbf{k}\] about the point \[2\,\mathbf{i}-\mathbf{j}+\mathbf{k}=\]                                  [BIT Ranchi 1992]
  A)
                  \[5\mathbf{i}-5\mathbf{j}+5\mathbf{k}\] done clear
  B)
                  \[5\mathbf{i}+5\mathbf{j}-5\mathbf{k}\] done clear
  C)
                  \[-5\mathbf{i}+5\mathbf{j}+5\mathbf{k}\] done clear
  D)
                  \[-5\mathbf{i}-5\mathbf{j}+5\mathbf{k}\] done clear
  View Solution play_arrow

• question_answer85)
  A force of magnitude 6 acts along the vector \[(9,\,6,\,-2)\] and passes through a point A (4, ? 1, ?7). The moment of the force about the point O (1, ? 3, 2) is
  A)
                  \[\frac{150}{11}\,(2\mathbf{i}-3\mathbf{j})\] done clear
  B)
                  \[\frac{6}{11}\,(50\mathbf{i}-75\mathbf{j}+36\mathbf{k})\] done clear
  C)
                  \[150\,(2\mathbf{i}-3\mathbf{j})\] done clear
  D)
                  \[6\,(50\mathbf{i}-75\mathbf{j}+36\mathbf{k})\] done clear
  View Solution play_arrow

• question_answer86)
  A force \[\mathbf{F}=2\mathbf{i}+\mathbf{j}-\mathbf{k}\] acts at a point A, whose position vector is \[2\mathbf{i}-\mathbf{j}\]. The moment of F about the origin is                                                                 [Karnataka CET 2000]
  A)
                  \[\mathbf{i}+2\mathbf{j}-4\mathbf{k}\] done clear
  B)
                  \[\mathbf{i}-2\mathbf{j}-4\mathbf{k}\] done clear
  C)
                  \[\mathbf{i}+2\mathbf{j}+4\mathbf{k}\] done clear
  D)
                  \[\mathbf{i}-2\mathbf{j}+4\mathbf{k}\] done clear
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• question_answer87)
  If \[\mathbf{a}=\mathbf{i}-\mathbf{j},\mathbf{b}=\mathbf{i}+\mathbf{j},\,\,\,\mathbf{c}=\mathbf{i}+3\mathbf{j}+5\mathbf{k}\]and\[\mathbf{n}\]is a unit vector such that \[\mathbf{b}.\mathbf{n}=0,\mathbf{a}.\mathbf{n}=0\]then the value of \[|\mathbf{c}\ .\ \mathbf{n}|\] is equal to [DCE 2005]
  A)
                  1 done clear
  B)
                  3 done clear
  C)
              5 done clear
  D)
                  2 done clear
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• question_answer88)
  A unit vector perpendicular to the plane containing the vectors \[\mathbf{i}-\mathbf{j}+\mathbf{k}\] and \[-\mathbf{i}+\mathbf{j}+\mathbf{k}\] is               [Karnataka CET 2005]
  A)
                  \[\frac{\mathbf{i}-\mathbf{j}}{\sqrt{2}}\] done clear
  B)
                  \[\frac{\mathbf{i}+\mathbf{k}}{\sqrt{2}}\] done clear
  C)
                  \[\frac{\mathbf{j}-\mathbf{k}}{\sqrt{2}}\] done clear
  D)
                  \[\frac{\mathbf{i}+\mathbf{j}}{\sqrt{2}}\] done clear
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