JEE Main & Advanced Mathematics Vector Algebra Question Bank

done Application of vectors in three dimensional geometry Total Questions - 47

• question_answer1)
  The locus of a point equidistant from two given points a and b is given by                 
  A)
             \[[\mathbf{r}-\frac{1}{2}(\mathbf{a}+\mathbf{b})]\,.\,\,(\mathbf{a}-\mathbf{b})=0\]   done clear
  B)
             \[[\mathbf{r}-\frac{1}{2}(\mathbf{a}-\mathbf{b})]\,.\,\,(\mathbf{a}+\mathbf{b})=0\] done clear
  C)
             \[[\mathbf{r}-\frac{1}{2}(\mathbf{a}+\mathbf{b})].(\mathbf{a}+\mathbf{b})=0\] done clear
  D)
             \[[\mathbf{r}-\frac{1}{2}(\mathbf{a}-\mathbf{b})]\,.\,\,(\mathbf{a}-\mathbf{b})=0\] done clear
  View Solution play_arrow

• question_answer2)
  If the non-zero vectors a and b are perpendicular to each other, then the solution of the equation \[\mathbf{r}\times \mathbf{a}=\mathbf{b}\] is given by
  A)
             \[\mathbf{r}=x\mathbf{a}+\frac{1}{\mathbf{a}\,.\,\,\mathbf{a}}(\mathbf{a}\times \mathbf{b})\] done clear
  B)
             \[\mathbf{r}=x\mathbf{b}-\frac{1}{\mathbf{b}\,.\,\,\mathbf{b}}(\mathbf{a}\times \mathbf{b})\] done clear
  C)
             \[\mathbf{r}=x\mathbf{a}\times \mathbf{b}\] done clear
  D)
             \[\mathbf{r}=x\mathbf{b}\times \mathbf{a}\] done clear
  View Solution play_arrow

• question_answer3)
  If r be position vector of any point on a sphere and a, b are respectively position vectors of the extremities of a diameter, then [MP PET 1994]
  A)
             \[\mathbf{r}\,.\,(\mathbf{a}-\mathbf{b})=0\] done clear
  B)
             \[\mathbf{r}\,.\,(\mathbf{r}-\mathbf{a})=0\] done clear
  C)
             \[(\mathbf{r}+\mathbf{a})\,.\,(\mathbf{r}+\mathbf{b})=0\] done clear
  D)
             \[(\mathbf{r}-\mathbf{a})\,.\,(\mathbf{r}-\mathbf{b})=0\] done clear
  View Solution play_arrow

• question_answer4)
  Angle between the line \[\mathbf{r}=(\mathbf{i}+2\mathbf{j}-\mathbf{k})+\lambda (\mathbf{i}-\mathbf{j}+\mathbf{k})\] and the normal to the plane \[\mathbf{r}\,.\,(2\mathbf{i}-\mathbf{j}+\mathbf{k})=4\] is [MP PET 1997]
  A)
             \[{{\sin }^{-1}}\,\left( \frac{2\sqrt{2}}{3} \right)\] done clear
  B)
             \[{{\cos }^{-1}}\,\left( \frac{2\sqrt{2}}{3} \right)\] done clear
  C)
             \[{{\tan }^{-1}}\,\left( \frac{2\sqrt{2}}{3} \right)\] done clear
  D)
             \[{{\cot }^{-1}}\,\left( \frac{2\sqrt{2}}{3} \right)\] done clear
  View Solution play_arrow

• question_answer5)
  If the equation of a line through a point a and parallel to vector b is \[\mathbf{r}=\mathbf{a}+t\,\mathbf{b},\] where t is a parameter, then its perpendicular distance from the point c is [MP PET 1998]
  A)
             \[|(\mathbf{c}-\mathbf{b})\times \mathbf{a}|\div |\mathbf{a}|\] done clear
  B)
             \[|(\mathbf{c}-\mathbf{a})\times \mathbf{b}|\div |\mathbf{b}|\] done clear
  C)
             \[|(\mathbf{a}-\mathbf{b})\times \mathbf{c}|\div |\mathbf{c}|\] done clear
  D)
             \[|(\mathbf{a}-\mathbf{b})\times \mathbf{c}|\div |\mathbf{a}+\mathbf{c}|\] done clear
  View Solution play_arrow

• question_answer6)
  If \[a=i+j\] and \[b=2i-k\] are two vectors, then the point of intersection of two lines \[r\times a=b\times a\]  and \[r\times b=a\times b\] is [RPET 2000]
  A)
             i + j ? k done clear
  B)
             i ? j + k done clear
  C)
             3i + j ? k done clear
  D)
             3i ? j + k done clear
  View Solution play_arrow

• question_answer7)
  If a, b, c are three non-coplanar vectors, then the vector equation \[\mathbf{r}=(1-\mathbf{p}-\mathbf{q})\,\mathbf{a}+p\mathbf{b}+q\mathbf{c}\] represents a             [EAMCET 2003]
  A)
             Straight line                               done clear
  B)
             Plane done clear
  C)
             Plane passing through the origin done clear
  D)
             Sphere done clear
  View Solution play_arrow

• question_answer8)
  The vector equation of the line joining the points \[i-2j+k\] and \[-2j+3k\] is [MP PET 2003]
  A)
             \[r=t(i+j+k)\]                           done clear
  B)
             \[r={{t}_{1}}(i-2j+k)+{{t}_{2}}(3k-2j)\] done clear
  C)
           \[r=(i-2j+k)+t(2k-i)\] done clear
  D)
             \[r=t(2k-i)\] done clear
  View Solution play_arrow

• question_answer9)
  The spheres\[{{r}^{2}}+2{{u}_{1}}\,.\,r+2{{d}_{1}}=0\] and \[{{r}^{2}}+2{{u}_{2}}\,.\,r+2{{d}_{2}}=0\] cut orthogonally, if [AMU 1999]
  A)
             \[{{u}_{1}}\,.\,{{u}_{2}}=0\]      done clear
  B)
             \[{{u}_{1}}+{{u}_{2}}=0\] done clear
  C)
             \[{{u}_{1}}\,.\,{{u}_{2}}={{d}_{1}}+{{d}_{2}}\] done clear
  D)
             \[({{u}_{1}}-{{u}_{2}})\,.\,({{u}_{1}}+{{u}_{2}})=d_{1}^{2}+d_{2}^{2}\] done clear
  View Solution play_arrow

• question_answer10)
  A tetrahedron has vertices at \[O(0,\,0,\,0)\], \[A(1,\,2,\,1),B(2,\,1,\,3)\] and \[C(-1,\,1,\,2)\]. Then the angle between the faces \[OAB\]and \[ABC\]will be        [MNR 1994; UPSEAT 2000; AIEEE 2003]
  A)
             \[{{\cos }^{-1}}\left( \frac{19}{35} \right)\] done clear
  B)
             \[{{\cos }^{-1}}\left( \frac{17}{31} \right)\] done clear
  C)
             \[30{}^\circ \]                          done clear
  D)
             \[90{}^\circ \] done clear
  View Solution play_arrow

• question_answer11)
  A vector \[\mathbf{n}\] of magnitude 8 units is inclined to x-axis at \[{{45}^{o}}\], y-axis at \[{{60}^{o}}\] and an acute angle with z-axis. If a plane passes through a point \[(\sqrt{2},\,-1,\,1)\] and is normal to \[\mathbf{n}\], then its equation in vector form is
  A)
             \[\mathbf{r}.(\sqrt{2}\mathbf{i}+\mathbf{j}+\mathbf{k})=4\] done clear
  B)
             \[\mathbf{r}.(\sqrt{2}\mathbf{i}+\mathbf{j}+\mathbf{k})=2\] done clear
  C)
             \[\mathbf{r}.(\mathbf{i}+\mathbf{j}+\mathbf{k})=4\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer12)
  The vector equation of a plane, which is at a distance of 8 unit from the origin and which is normal to the vector \[2\mathbf{i}+\mathbf{j}+2\mathbf{k},\] is
  A)
             \[\mathbf{r}.(2\mathbf{i}+\mathbf{j}+\mathbf{k})=24\] done clear
  B)
             \[\mathbf{r}.(2\mathbf{i}+\mathbf{j}+2\mathbf{k})=24\] done clear
  C)
             \[\mathbf{r}.(\mathbf{i}+\mathbf{j}+\mathbf{k})=24\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer13)
  The distance of the point \[2\mathbf{i}+\mathbf{j}-\mathbf{k}\] from the plane \[\mathbf{r}.(\mathbf{i}-2\mathbf{j}+4\mathbf{k})=9\] is         
  A)
             \[\frac{13}{\sqrt{21}}\] done clear
  B)
             \[\frac{13}{21}\]                      done clear
  C)
             \[\frac{13}{3\sqrt{21}}\] done clear
  View Solution play_arrow

• question_answer14)
  The centre of the circle given by \[\mathbf{r}.(\mathbf{i}+2\mathbf{j}+2\mathbf{k})=15\] and \[|\mathbf{r}-(\mathbf{j}+2\mathbf{k})|=4\]is   
  A)
             (0, 1, 2) done clear
  B)
             (1, 3, 4) done clear
  C)
             (?1, 3, 4) done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer15)
  A vector r is equally inclined with the co-ordinate axes. If the tip of r is in the positive octant and |r| = 6, then \[\mathbf{r}\] is
  A)
             \[2\sqrt{3}(\mathbf{i}-\mathbf{j}+\mathbf{k})\] done clear
  B)
             \[2\sqrt{3}(-\mathbf{i}+\mathbf{j}+\mathbf{k})\] done clear
  C)
             \[2\sqrt{3}(\mathbf{i}+\mathbf{j}-\mathbf{k})\] done clear
  D)
             \[2\sqrt{3}(\mathbf{i}+\mathbf{j}+\mathbf{k})\] done clear
  View Solution play_arrow

• question_answer16)
  The position vectors of two points P and Q are \[3\mathbf{i}+\mathbf{j}+2\mathbf{k}\] and \[\mathbf{i}-2\mathbf{j}-4\mathbf{k}\] respectively. The equation of the plane through Q and perpendicular to PQ is
  A)
             \[\mathbf{r}.(2\mathbf{i}+3\mathbf{j}+6\mathbf{k})=28\] done clear
  B)
             \[\mathbf{r}.(2\mathbf{i}+3\mathbf{j}+6\mathbf{k})=32\] done clear
  C)
             \[\mathbf{r}.(2\mathbf{i}+3\mathbf{j}+6\mathbf{k})+28=0\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer17)
  The vector equation of the plane passing through the origin and the line of intersection of the plane \[\mathbf{r}.\mathbf{a}=\lambda \] and \[\mathbf{r}.\mathbf{b}=\mu \] is
  A)
             \[\mathbf{r}.(\lambda \mathbf{a}-\mu \mathbf{b})=0\] done clear
  B)
             \[\mathbf{r}.\,(\lambda \mathbf{b}-\mu \mathbf{a})=0\] done clear
  C)
             \[\mathbf{r}.(\lambda \mathbf{a}+\mu \mathbf{b})=0\] done clear
  D)
             \[\mathbf{r}.(\lambda \mathbf{b}+\mu \mathbf{a})=0\] done clear
  View Solution play_arrow

• question_answer18)
  The position vectors of points A and B are \[\mathbf{i}-\mathbf{j}+3\mathbf{k}\] and \[3\mathbf{i}+3\mathbf{j}+3\mathbf{k}\] respectively. The equation of a plane is \[\mathbf{r}.(5\mathbf{i}+2\mathbf{j}-7\mathbf{k})+9=0\]. The points A and B
  A)
             Lie on the plane                       done clear
  B)
             Are on the same side of the plane done clear
  C)
             Are on the opposite side of the plane done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer19)
  The vector equation of the plane through the point \[2\mathbf{i}-\mathbf{j}-4\mathbf{k}\] and parallel to the plane \[\mathbf{r}.(4\mathbf{i}-12\mathbf{j}-3\mathbf{k})-7=0\] is           
  A)
             \[\mathbf{r}.(4\mathbf{i}-12\mathbf{j}-3\mathbf{k})=0\] done clear
  B)
             \[\mathbf{r}.(4\mathbf{i}-12\mathbf{j}-3\mathbf{k})=32\] done clear
  C)
             \[\mathbf{r}.(4\mathbf{i}-12\mathbf{j}-3\mathbf{k})=12\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer20)
  The vector equation of the plane through the point (2, 1, ?1) and passing through the line of intersection of the plane \[\mathbf{r}.(\mathbf{i}+3\mathbf{j}-\mathbf{k})=0\] and \[\mathbf{r}.(\mathbf{j}+2\mathbf{k})=0\] is
  A)
             \[\mathbf{r}.(\mathbf{i}+9\mathbf{j}+11\mathbf{k})=0\] done clear
  B)
             \[\mathbf{r}.(\mathbf{i}+9\mathbf{j}+11\mathbf{k})=6\] done clear
  C)
             \[\mathbf{r}.(\mathbf{i}-3\mathbf{j}-13\mathbf{k})=0\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer21)
  The vector equation of the plane through the point \[\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and perpendicular to the line of intersection of the planes \[\mathbf{r}.(3\mathbf{i}-\mathbf{j}+\mathbf{k})=1\]  and \[\mathbf{i}+4\mathbf{j}-2\mathbf{k}=2\] is
  A)
             \[\mathbf{r}.(2\mathbf{i}+7\mathbf{j}-13\mathbf{k})=1\] done clear
  B)
             \[\mathbf{r}.(2\mathbf{i}-7\mathbf{j}-13\mathbf{k})=1\] done clear
  C)
             \[\mathbf{r}.(2\mathbf{i}+7\mathbf{j}+13\mathbf{k})=0\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer22)
  The equation of the plane containing the lines \[\mathbf{r}={{\mathbf{a}}_{1}}+\lambda {{\mathbf{a}}_{2}}\] and \[\mathbf{r}={{\mathbf{a}}_{2}}+\lambda {{\mathbf{a}}_{1}}\] is
  A)
           \[[\mathbf{r}\,\ {{\mathbf{a}}_{1}}\ \,{{\mathbf{a}}_{2}}]=0\] done clear
  B)
             \[[\mathbf{r}\ \,{{\mathbf{a}}_{1}}\ \,{{\mathbf{a}}_{2}}]={{\mathbf{a}}_{1}}.\ {{\mathbf{a}}_{2}}\] done clear
  C)
             \[[\mathbf{r}\ \,{{\mathbf{a}}_{2}}\ \,{{\mathbf{a}}_{1}}]={{\mathbf{a}}_{1}}.\ {{\mathbf{a}}_{2}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer23)
  The vector equation of the plane containing the lines \[\mathbf{r}=(\mathbf{i}+\mathbf{j})+\lambda (\mathbf{i}+2\mathbf{j}-\mathbf{k})\] and \[\mathbf{r}=(\mathbf{i}+\mathbf{j})+\mu (-\mathbf{i}+\mathbf{j}-2\mathbf{k})\] is
  A)
             \[\mathbf{r}.(\mathbf{i}+\mathbf{j}+\mathbf{k})=0\] done clear
  B)
             \[\mathbf{r}.(\mathbf{i}-\mathbf{j}-\mathbf{k})=0\] done clear
  C)
             \[\mathbf{r}.(\mathbf{i}+\mathbf{j}+\mathbf{k})=3\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer24)
  The cartesian equation of the plane          \[\mathbf{r}=(1+\lambda -\mu )\mathbf{i}+(2-\lambda )\mathbf{j}+(3-2\lambda +2\mu )\mathbf{k}\] is           
  A)
             \[2x+y=5\] done clear
  B)
             \[2x-y=5\] done clear
  C)
             \[2x+z=5\] done clear
  D)
             \[2x-z=5\] done clear
  View Solution play_arrow

• question_answer25)
  The length of the perpendicular from the origin to the plane passing through three non-collinear points \[\mathbf{a},\,\mathbf{b},\,\mathbf{c}\] is
  A)
           \[\frac{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}{|\mathbf{a}\times \mathbf{b}+\mathbf{c}\times \mathbf{a}+\mathbf{b}\times \mathbf{c}|}\] done clear
  B)
             \[\frac{2\,[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}{|\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a}|}\] done clear
  C)
             \[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer26)
  The length of the perpendicular from the origin to the plane passing through the point a and containing the line \[\mathbf{r}=\mathbf{b}+\lambda \mathbf{c}\] is
  A)
             \[\frac{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}{|\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a}|}\] done clear
  B)
             \[\frac{\,[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}{|\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}|}\] done clear
  C)
           \[\frac{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}{|\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a}|}\] done clear
  D)
             \[\frac{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}{|\mathbf{c}\times \mathbf{a}+\mathbf{a}\times \mathbf{b}|}\] done clear
  View Solution play_arrow

• question_answer27)
  The position vector of a point at a distance of \[3\sqrt{11}\] units from \[\mathbf{i}-\mathbf{j}+2\mathbf{k}\] on a line passing through the points \[\mathbf{i}-\mathbf{j}+2\mathbf{k}\] and \[3\mathbf{i}+\mathbf{j}+\mathbf{k}\] is                                
  A)
             \[10\mathbf{i}+2\mathbf{j}-5\mathbf{k}\] done clear
  B)
             \[-8\mathbf{i}-4\mathbf{j}-\mathbf{k}\] done clear
  C)
             \[8\mathbf{i}+4\mathbf{j}+\mathbf{k}\] done clear
  D)
             \[-10\mathbf{i}-2\mathbf{j}-5\mathbf{k}\] done clear
  View Solution play_arrow

• question_answer28)
  The line joining the points \[6\mathbf{a}-4\mathbf{b}+4\mathbf{c},\,-4\mathbf{c}\] and the line joining the points \[-\mathbf{a}-2\mathbf{b}-3\mathbf{c},\,\mathbf{a}+2\mathbf{b}-5\mathbf{c}\] intersect at
  A)
             \[-4\mathbf{a}\] done clear
  B)
             \[4\mathbf{a}-\mathbf{b}-\mathbf{c}\] done clear
  C)
             \[4\mathbf{c}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer29)
  Angle between the line \[\mathbf{r}=(2\mathbf{i}-\mathbf{j}+\mathbf{k})+\lambda (-\mathbf{i}+\mathbf{j}+\mathbf{k})\] and the plane \[\mathbf{r}.(3\mathbf{i}+2\mathbf{j}-\mathbf{k})=4\] is
  A)
             \[{{\cos }^{-1}}\left( \frac{2}{\sqrt{42}} \right)\] done clear
  B)
             \[{{\cos }^{-1}}\left( \frac{-2}{\sqrt{42}} \right)\] done clear
  C)
             \[{{\sin }^{-1}}\left( \frac{2}{\sqrt{42}} \right)\] done clear
  D)
             \[{{\sin }^{-1}}\left( \frac{-2}{\sqrt{42}} \right)\] done clear
  View Solution play_arrow

• question_answer30)
  The line through \[\mathbf{i}+3\mathbf{j}+2\mathbf{k}\] and perpendicular to the lines \[\mathbf{r}=(\mathbf{i}+2\mathbf{j}-\mathbf{k})+\lambda (2\mathbf{i}+\mathbf{j}+\mathbf{k})\]             and \[\mathbf{r}=(2\mathbf{i}+6\mathbf{j}+\mathbf{k})+\mu (\mathbf{i}+2\mathbf{j}+3\mathbf{k})\] is
  A)
             \[\mathbf{r}=(\mathbf{i}+2\mathbf{j}-\mathbf{k})+\lambda (-\mathbf{i}+5\mathbf{j}-3\mathbf{k})\] done clear
  B)
             \[\mathbf{r}=\mathbf{i}+3\mathbf{j}+2\mathbf{k}+\lambda (\mathbf{i}-5\mathbf{j}+3\mathbf{k})\] done clear
  C)
             \[\mathbf{r}=\mathbf{i}+3\mathbf{j}+2\mathbf{k}+\lambda (\mathbf{i}+5\mathbf{j}+3\mathbf{k})\] done clear
  D)
             \[\mathbf{r}=\mathbf{i}+3\mathbf{j}+2\mathbf{k}+\lambda (-\mathbf{i}+5\mathbf{j}-3\mathbf{k})\] done clear
  View Solution play_arrow

• question_answer31)
  The distance from the point \[-\mathbf{i}+2\mathbf{j}+6\mathbf{k}\] to the straight line through the point (2, 3, ?4) and parallel to the vector \[6\mathbf{i}+3\mathbf{j}-4\mathbf{k}\] is
  A)
             7 done clear
  B)
             10 done clear
  C)
             9 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer32)
  The position vector of the point in which the line joining the points \[\mathbf{i}-2\mathbf{j}+\mathbf{k}\] and \[3\mathbf{k}-2\mathbf{j}\] cuts the plane through the origin and the points \[4\mathbf{j}\] and \[2\mathbf{i}+\mathbf{k}\], is
  A)
             \[6\mathbf{i}-10\mathbf{j}+3\mathbf{k}\] done clear
  B)
           \[\frac{1}{5}(6\mathbf{i}-10\mathbf{j}+3\mathbf{k})\] done clear
  C)
             \[-6\mathbf{i}+10\mathbf{j}-3\mathbf{k}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer33)
  The distance between the planes given by \[\mathbf{r}.(\mathbf{i}+2\mathbf{j}-2\mathbf{k})+5=0\] and \[\mathbf{r}.(\mathbf{i}+2\mathbf{j}-2\mathbf{k})-8=0\] is
  A)
             1 unit done clear
  B)
             \[\frac{13}{3}\] unit done clear
  C)
             13 unit done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer34)
  The equation of the plane containing the line \[\mathbf{r}=\mathbf{i}+\mathbf{j}+\lambda (2\mathbf{i}+\mathbf{j}+4\mathbf{k})\] is
  A)
             \[\mathbf{r}.(\mathbf{i}+2\mathbf{j}-\mathbf{k})=3\] done clear
  B)
             \[\mathbf{r}.(\mathbf{i}+2\mathbf{j}-\mathbf{k})=6\] done clear
  C)
             \[\mathbf{r}.(-\mathbf{i}-2\mathbf{j}+\mathbf{k})=3\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer35)
  The equation \[|\mathbf{r}{{|}^{2}}-\mathbf{r}.(2\mathbf{i}+4\mathbf{j}-2\mathbf{k})-10=0\] represents a
  A)
             Circle done clear
  B)
             Plane done clear
  C)
             Sphere of radius 4 done clear
  D)
             Sphere of radius 3 done clear
  E)
             None of these done clear
  View Solution play_arrow

• question_answer36)
  The centre of the sphere \[\alpha \,\mathbf{r}-2\mathbf{u}.\mathbf{r}=\beta ,(\alpha \ne 0)\] is [AMU 1999]
  A)
             \[-\mathbf{u}/\alpha \] done clear
  B)
             \[\mathbf{u}/\alpha \] done clear
  C)
             \[\alpha \mathbf{u}/\beta \] done clear
  D)
             \[\frac{\alpha +\beta }{\alpha }\mathbf{u}\] done clear
  View Solution play_arrow

• question_answer37)
  The shortest distance between the lines \[\mathbf{r}=(3\mathbf{i}-2\mathbf{j}-2\mathbf{k})+\mathbf{i}t\] and \[\mathbf{r}=\mathbf{i}-\mathbf{j}+2\mathbf{k}+\mathbf{j}s\] (t and s being parameters) is           [AMU 1999]
  A)
             \[\sqrt{21}\] done clear
  B)
             \[\sqrt{102}\] done clear
  C)
             4 done clear
  D)
             3 done clear
  View Solution play_arrow

• question_answer38)
  The equation of the line passing through the points \[{{a}_{1}}\mathbf{i}+{{a}_{2}}\mathbf{j}+{{a}_{3}}\mathbf{k}\] and \[{{b}_{1}}\mathbf{i}+{{b}_{2}}\mathbf{j}+{{b}_{3}}\mathbf{k}\]is   [RPET 2002]
  A)
  \[({{a}_{1}}\mathbf{i}+{{a}_{2}}\mathbf{j}+{{a}_{3}}\mathbf{k})+t({{b}_{1}}\mathbf{i}+{{b}_{2}}\mathbf{j}+{{b}_{3}}\mathbf{k})\] done clear
  B)
             \[({{a}_{1}}\mathbf{i}+{{a}_{2}}\mathbf{j}+{{a}_{3}}\mathbf{k})-t({{b}_{1}}\mathbf{i}+{{b}_{2}}\mathbf{j}+{{b}_{3}}\mathbf{k})\] done clear
  C)
           \[{{a}_{1}}(1-t)\mathbf{i}+{{a}_{2}}(1-t)\mathbf{j}+{{a}_{3}}(1-t)\mathbf{k}+({{b}_{1}}\mathbf{i}+{{b}_{2}}\mathbf{j}+{{b}_{3}}\mathbf{k})t\] done clear
  D)
           None of these done clear
  View Solution play_arrow

• question_answer39)
  The distance between the line \[\mathbf{r}=2\mathbf{i}-2\mathbf{j}+3\mathbf{k}+\lambda (\mathbf{i}-\mathbf{j}+4\mathbf{k})\] and the plane \[\mathbf{r}.(\mathbf{i}+5\mathbf{j}+\mathbf{k})=5\] is [AIEEE 2005]
  A)
             \[\frac{3}{10}\] done clear
  B)
             \[\frac{10}{3}\] done clear
  C)
             \[\frac{10}{9}\] done clear
  D)
             \[\frac{10}{3\sqrt{3}}\] done clear
  View Solution play_arrow

• question_answer40)
  The image of the point with position vector \[\mathbf{i}+3\mathbf{k}\]in the plane \[\mathbf{r}.(\mathbf{i}+\mathbf{j}+\mathbf{k})=1\]is [J & K 2005]
  A)
             \[\mathbf{i}+2\mathbf{j}+\mathbf{k}\] done clear
  B)
             \[\mathbf{i}-2\mathbf{i}+\mathbf{k}\] done clear
  C)
             \[-\mathbf{i}-2\mathbf{j}+\mathbf{k}\] done clear
  D)
             \[\mathbf{i}+2\mathbf{j}-\mathbf{k}\] done clear
  View Solution play_arrow

• question_answer41)
  The equation of the plane passing through the points \[(-1,-2,\,0),(2,\,3,\,5)\] and parallel to the line                  \[\mathbf{r}=-3\mathbf{j}+\mathbf{k}+\mathbf{\lambda }(2\mathbf{i}+5\mathbf{j}-\mathbf{k})\] is [J & K 2005]
  A)
             \[\mathbf{r}.(-30\mathbf{i}+13\mathbf{j}+5\mathbf{k})=4\] done clear
  B)
             \[\mathbf{r}.(30\mathbf{i}+13\mathbf{j}+5\mathbf{k})=4\] done clear
  C)
             \[\mathbf{r}.(30\mathbf{i}+13\mathbf{j}-5\mathbf{k})=4\] done clear
  D)
             \[\mathbf{r}.(30\mathbf{i}-13\mathbf{j}-5\mathbf{k})=4\] done clear
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• question_answer42)
  The shortest distance between the lines \[{{\mathbf{r}}_{1}}=4\mathbf{i}-3\mathbf{j}-\mathbf{k}+\lambda (\mathbf{i}-4\mathbf{j}+7\mathbf{k})\]            and \[{{\mathbf{r}}_{2}}=\mathbf{i}-\mathbf{j}-10\mathbf{k}+\lambda (2\mathbf{i}-3\mathbf{j}+8\mathbf{k})\]is [J & K 2005]
  A)
             3 done clear
  B)
             1 done clear
  C)
             2 done clear
  D)
           0 done clear
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• question_answer43)
  The position vector of the point where the line \[\mathbf{r}=\mathbf{i}-\mathbf{j}+\mathbf{k}+t(\mathbf{i}+\mathbf{j}+\mathbf{k})\]meets the plane \[\mathbf{r}.(\mathbf{i}+\mathbf{j}+\mathbf{k})=5\]is [Kerala (Engg.) 2005]
  A)
             \[5\mathbf{i}+\mathbf{j}-\mathbf{k}\] done clear
  B)
             \[5\mathbf{i}+3\mathbf{j}-3\mathbf{k}\] done clear
  C)
             \[2\mathbf{i}+\mathbf{j}+2\mathbf{k}\] done clear
  D)
             \[5\mathbf{i}+\mathbf{j}+\mathbf{k}\] done clear
  E)
             \[4\mathbf{i}+2\mathbf{j}-2\mathbf{k}\] done clear
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• question_answer44)
  A Plane meets the co-ordinate axes at P, Q and R such that the position vector of the centroid of \[\Delta PQR\] is \[2\mathbf{i}-5\mathbf{j}+8\mathbf{k}\]. Then the equation of the plane is [J & K 2005] 
  A)
             \[\mathbf{r}.(20\mathbf{i}-8\mathbf{j}+5\mathbf{k})=120\] done clear
  B)
             \[\mathbf{r}.(20\mathbf{i}-8\mathbf{j}+5\mathbf{k})=1\] done clear
  C)
             \[\mathbf{r}.(20\mathbf{i}-8\mathbf{j}+5\mathbf{k})=2\] done clear
  D)
             \[\mathbf{r}.(20\mathbf{i}-8\mathbf{j}+5\mathbf{k})=20\] done clear
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• question_answer45)
  The line of intersection of the planes \[\mathbf{r}.(\mathbf{i}-3\mathbf{j}+\mathbf{k})=1\] and \[\mathbf{r}.(2\mathbf{i}+5\mathbf{j}-3\mathbf{k})=2\] is parallel to the vector 
  A)
             \[-4\mathbf{i}+5\mathbf{j}+11\mathbf{k}\] done clear
  B)
             \[4\mathbf{i}+5\mathbf{j}+11\mathbf{k}\] done clear
  C)
             \[4\mathbf{i}-5\mathbf{j}+11\mathbf{k}\] done clear
  D)
             \[4\mathbf{i}-5\mathbf{j}-11\mathbf{k}\] done clear
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• question_answer46)
  The equation of plane passing through a point \[A(2,-1,\,3)\] and parallel to the vectors \[\mathbf{a}=(3,\,0,-1)\] and \[\mathbf{b}=(-3,\,\,2,\,2)\] is                                                                       [Orissa JEE 2005]
  A)
             \[2x-3y+6z-25=0\] done clear
  B)
             \[2x-3y+6z+25=0\] done clear
  C)
             \[3x-2y+6z-25=0\] done clear
  D)
             \[3x-2y+6z+25=0\] done clear
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• question_answer47)
  If the position vectors of two point P and Q are respectively \[9\mathbf{i}-\mathbf{j}+5\mathbf{k}\] and \[\mathbf{i}+3\mathbf{j}+5\mathbf{k}\], and the line segment PQ intersects the YOZ plane at a point R, the PR : RQ is equal to                                                                           [J & K 2005]
  A)
             9 : 1 done clear
  B)
             1 : 9 done clear
  C)
        ?1 : 9 done clear
  D)
       ? 9 : 1 done clear
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