11th Class Mathematics Three Dimensional Geometry-300 Notes - Mathematics Olympiads - Three Dimensional Plane

Notes - Mathematics Olympiads - Three Dimensional Plane

Category : 11th Class


                                                                           Three Dimensional Plane


In three dimensional Geometry, it is not a new geometry though it is the refined or extension form of the two dimension geometry. In 3-dimensional geometry. Three axes i.e. x-axis, y-axis and z-axis are perpendicular to each other is considered.

Let \[X'OX',Y'OY\] & \[Z'OZ\] be three mutually perpendicular lines which be intersect at 0. It is called origin.









Plane XOY is called xy plane

YOZ is called yz plane

and ZOX is called zx plane

In 3-D, there are 8 quadrents

Equation of x-axis be y= 0 & z =0

Equation of y-axis be x = 0 & z = 0

and equation of z-axis be x=0 & y=0


Note: In 3-D, a straight line is represented by two equations where as a plane is represented by single equation in at most three variables.


  • Some basic formula which are used in 3-dimension. The distance between points \[A({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})\] and \[B({{x}_{1}},\,{{y}_{2}},\,{{z}_{3}})\]be




e.g. Let two points are A (2, 3, 1) & B = (- 5, 2-1)


            \[\therefore \,\,\,\,AB=\sqrt{{{(-5-2)}^{2}}+{{(2-3)}^{2}}+{{(-1-1)}^{2}}}\]




  • Section Formula: The coordinate of the point P dividing the line joining \[A({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})\] & \[({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})\] in the ratio m:n internally are


\[P=\left( \frac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\frac{m{{y}_{2}}+n{{y}_{1}}}{m+n},\frac{m{{z}_{2}}+n{{z}_{1}}}{m+n} \right)\]


The co-ordinate of the point P dividing the line joining \[A({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})\] and \[({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})\] in the ratio m:n externally are


\[P=\left( \frac{m{{x}_{2}}-n{{x}_{1}}}{m-n},\frac{m{{y}_{2}}-n{{y}_{1}}}{m-n},\frac{m{{z}_{2}}-n{{z}_{1}}}{m-n} \right)\]


Midpoint of AB be


            \[P=\left( \frac{{{x}_{1}}+{{x}_{2}}}{2},\frac{{{y}_{1}}+{{y}_{2}}}{2},\frac{{{z}_{1}}+{{z}_{2}}}{2} \right).\]


e.g.       Find the co-ordinate of the point which divides the line segment joining the point (-2, 3, 5) & (1, - 4, 6) in the ratio (i) 2: 3 internally (ii) 2:3 externally.


Sol.      Here, Let A= (-2, 3, 5) & B= (1, -4, 6)

and m:n =2:3 internally

Let P divides AB in the ratio m: n internally


            \[\therefore \,\,\,P=\left( \frac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\frac{m{{y}_{2}}+n{{y}_{1}}}{m+n},\frac{m{{z}_{2}}+m{{z}_{1}}}{m+n} \right)\]


            \[=\left( \frac{2.1+3(-2)}{2+3},\frac{2(-4)+3.3}{2+3},\frac{2\times 6-3\times 5}{2+3} \right)\]


            \[=\left( \frac{-4}{5},\frac{1}{5},\frac{27}{5} \right)\]


When P divides AB in the ratio m : n externally


            \[\therefore \,\,\,P=\left( \frac{2.1-3(-2)}{2-3},\frac{2(-4)-3.3}{2-3},\frac{2\times 6-3\times 5}{2-3} \right)\]




  • Controid of triangle: The co-ordinate of the centroid of the triangle ABC, whose vertices are


\[A({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}}),\] \[B({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}}),\] & \[C({{x}_{2}},\,{{y}_{2}},\,{{z}_{3}}),\] are


\[\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\frac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3},\frac{{{z}_{1}}+{{z}_{2}}+{{z}_{3}}}{3} \right)\]


  • Centroid of the tetrahedran: If \[({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}}),\] \[({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}}),\] \[({{x}_{3}},\,{{y}_{3}},\,{{z}_{3}})\,a({{x}_{4}},\,{{y}_{4}},\,{{z}_{4}})\] be the vertices of the tetrahedran, then its centroid G is given by


\[\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}}{4},\frac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}+{{y}_{4}}}{4},\frac{z1+z2+z3+z4}{4} \right)\]


  1. A point R with x-coordinate 4 lies on the segment joining the points \[\mathbf{P(2,-3,4)}\] & \[\mathbf{Q(8,0,1,0)}\mathbf{.}\]

Find the co-ordinate of the point R.


Sol:      Let \[P(2,-3,4)\] and \[Q(8,0,1,0).\]

Let R divides PQ in the ratio l: 1 internally


\[\therefore \,\,\,\,R=\left( \frac{8\lambda +2}{\lambda +1},\frac{0.\lambda +(-3)}{\lambda +1},\frac{10\lambda +1\times 4}{\lambda +1} \right)\]…………  (1)

Here x co-ordinate =4


\[\therefore \,\,\,\,\frac{8\lambda +2}{\lambda +1}=\frac{4}{1}\]


\[\Rightarrow \,\,8\lambda +2=4\lambda +4\]

\[\Rightarrow \,\,8\lambda -4\lambda =4-2=2\]

\[\Rightarrow \,\,4\lambda =2\]

\[\lambda =\frac{2}{4}=\frac{1}{2}\]


Putting the value of \[\lambda \] in (1), we have


\[R=\left( \frac{4+2}{\frac{1}{2}+1},\frac{-3}{\frac{1}{2}+1},\frac{5+4}{\frac{1}{2}+1}, \right)\]


\[=\left( 6\times \frac{2}{3},-3\times \frac{2}{3},9\times \frac{2}{3} \right)=(4,-2,6)\]


  1. Find the co-ordinate of the point R on -y axis which are at distance of \[\mathbf{5}\sqrt{\mathbf{2}}\] from the point \[\mathbf{P(3,-2,5)}\]


Sol:      Let \[R=(0,\,b,\,0)\] on y-axis.

Given \[P(3,-2,5)\] & \[PR=5\sqrt{2}\]

By distance formula,




squaring both side, we have


            \[9+{{(2+b)}^{2}}+25=25\times 2\]


\[\Rightarrow {{b}^{2}}+4b+38-50=0\]


\[\Rightarrow b=2,-6\]

Hence b =2 (b=-6 is not possible)

            \[\therefore \,\,\,R=(0,2,0)\]


  • Direction cosines: Let P (a, b, c) be any point. We join P to origin O. Let the line OP makes an angle \[\alpha \,,\beta ,\,\gamma \] with positive direction of x-axis, y-axis and z-axis respectively. Then \[\cos \,\,\alpha \,,\] \[\cos \,\,\beta ,\]and \[\cos \,\,\gamma \]and cosy are called the direction cosine of the directed line OP. If the angle is measured in clockwise direction then the direction angles are replaced by their supplements i.e. \[\pi -\alpha ,\] \[\pi -\beta ,\] and \[\pi -\gamma \] respectively. It is generally denoted by t, m and n respectively i.e. \[\ell =\cos \alpha ,\,\,\,\,m\]

\[=\cos \beta \] and \[n=\cos \,\,\gamma \]

\[\Rightarrow \,\,\,{{\ell }^{2}}+{{m}^{2}}+{{n}^{2}}=1\]


Direction Ratio: The three number a, b, c proportional to the direction cosines \[\ell \], m, n of a vector are known as the direction ratio of the OP vector,



We consider P (a, b, c) be any point in the space length f from the origin to the axis is said to be direction ratio.


\[\therefore \,\,\,\,OP=\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}\] (By distance formulae)


\[\left| \,r\, \right|=\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}\]


Note: Direction cosine is proportional to the direction ratio. Let a, b, c be d.r. of the line OP and its d.c. be \[\ell \], m and n respectively.


Then \[\frac{\ell }{a}=\frac{m}{b}=\frac{n}{c}=K\] (say)


Convection from d.r. to D.C.


\[\ell =\pm \frac{a}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\]


            \[m=\pm \frac{b}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\]


            and \[n=\pm \frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\]


  • Some silent features about D.R. and D.C.


If \[\overline{r}=a\overline{i}+b\overline{j}+c\overline{z}\]

Then a, b and c be the d.r. of r and d.c. of r be


            \[\ell =\frac{a}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}},m=\frac{b}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\] and \[n=\frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\]


  • The D.R. of the line joining two point \[P({{x}_{1}},{{y}_{1}},{{z}_{1}})\] and \[Q({{x}_{2}},{{y}_{2}},{{z}_{2}})\] are \[{{x}_{2}}-{{x}_{1}},\,\]\[{{y}_{2}}-{{y}_{1}}\] and \[{{z}_{2}}-{{z}_{1}}\] and its direction cosine (d.c.) be


\[\frac{{{x}_{2}}-{{x}_{1}}}{\left| PQ \right|},\frac{{{y}_{2}}-{{y}_{1}}}{\left| PQ \right|},\frac{{{z}_{2}}-{{z}_{1}}}{\left| PQ \right|}\] respectively


  • Direction cosine of x-axis, y-axis and z-axis be written as (1, 0, 0) (0, 1, 0) and (0, 0, 1) respectively


  • Angle between Two Vectors


If \[\theta \] be the angle between two vectors whose direction cosines are \[{{\ell }_{1}},\,{{m}_{1}},\,{{n}_{1}}\]and \[{{\ell }_{2}},\,{{m}_{2}},\,{{n}_{2}}\] then

\[\cos \theta ={{\ell }_{1}}{{\ell }_{2}}+{{m}_{1}}{{m}_{2}}+{{n}_{1}}{{n}_{2}}\]and


            \[\sin \theta =\sqrt{{{({{m}_{1}}{{n}_{2}}-{{m}_{2}}{{n}_{1}})}^{2}}+{{({{m}_{1}}{{\ell }_{2}}-{{m}_{2}}{{\ell }_{1}})}^{2}}+{{({{\ell }_{1}}{{n}_{2}}-{{\ell }_{2}}{{n}_{1}})}^{2}}}\]


  • If \[{{\ell }_{1}}{{\ell }_{2}}+{{m}_{1}}{{m}_{2}}+{{n}_{1}}{{n}_{2}}=0\]

Then both vection be orthogonal.


  • If \[\frac{{{\ell }_{1}}}{{{\ell }_{2}}}=\frac{{{m}_{1}}}{{{m}_{2}}}=\frac{{{n}_{1}}}{{{n}_{2}}}\]

Then both vectors are parallel.


  • Angle in the terms of direction ratio (D.R).

If \[\overline{a}={{a}_{1}}\overline{i}+{{b}_{1}}\overline{j}+{{c}_{1}}\overline{k}\]

and \[\overline{b}={{a}_{2}}\overline{i}+{{b}_{2}}\overline{j}+{{c}_{2}}\overline{k}\] be two vectors


\[\therefore \]      Its direction ratios be \[{{a}_{1}},\,{{b}_{1}},\,{{c}_{1}}\] and \[{{a}_{2}},\,{{b}_{2}},\,{{c}_{2}}\] respectively, q is the angle between these two vectors. Then


            \[\therefore \,\,\,\,\,\,\cos \theta =\frac{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}}{\sqrt{{{a}^{2}}_{1}+{{b}^{2}}_{1}+{{c}^{2}}_{1}}.\sqrt{{{a}^{2}}_{2}+{{b}^{2}}_{2}+{{c}^{2}}_{2}}}.\]


If two vectors are orthogonal then




If two vectors are parallel then \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{{{c}_{1}}}{{{c}_{2}}}\]


  • Projection of the joining of the two points on a line: If \[P({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})\] and \[Q({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})\] are two points, then the length of the projection of PQ on a line whose direction cosine are \[\ell \], m & n is written

as \[({{x}_{2}}-{{x}_{1}}).\ell +({{y}_{2}}-{{y}_{1}}).m+({{z}_{2}}-{{z}_{1}}).n\]


  1. Find the projection of the line segment joining the points (1,1,2) and (3,4,1) on the line whose direction ratio be (2,3, 6)


Sol.      Here P= (1, 1, 2), Q= (3, 4, 1)

Given d.r. = (2, 3, 6)

1st of all we have to find the d.c.


            \[\therefore \,\,\,\,\,\,\ell =\frac{2}{\sqrt{{{2}^{2}}+{{3}^{2}}+{{6}^{2}}}}=\frac{2}{7}\]


            \[m=\frac{3}{4}\] and \[n=\frac{6}{7}\]

\[\therefore \] Projection of the line joining the points (1, 1, 2) and (3, 4, 1) an the line whose d.c. be \[\left( \frac{2}{7},\frac{3}{7},\frac{6}{7} \right)\] is written as

            \[=(3-1)\times \frac{2}{7}+(4-1)\times \frac{3}{7}+(1-2)\times \frac{6}{7}=2\times \frac{2}{7}+3\times \frac{3}{7}+\frac{-6}{7}\]



Notes - Mathematics Olympiads - Three Dimensional Plane
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