CLASS XII MATHEMATICS CHAPTER 11

Three Dimensional Geometry

Complete Class XII Mathematics chapter notes covering vectors in three dimensions, direction cosines, direction ratios, equations of lines and planes, angles, shortest distance and important geometrical applications.

CBSE Pattern 3D Coordinate System Lines Planes Shortest Distance HOT Questions

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Chapter Overview

Introduction

Three Dimensional Geometry deals with the geometry of points, lines and planes in space. Unlike plane geometry, a point in space requires three coordinates \(x,y,z\).

The chapter connects algebra with spatial geometry through direction ratios, direction cosines, vector equations, Cartesian equations of lines and planes, angles and distances.

\[ P(x,y,z) \quad\Longleftrightarrow\quad \vec r=x\hat i+y\hat j+z\hat k \]

Spatial Coordinates

Three Dimensional Coordinate System

Coordinates of a Point

A point \(P\) in space is represented as

\[ P(x,y,z) \]

Here \(x\), \(y\), and \(z\) are the directed distances from the coordinate planes.

Fundamental Formulae

Distance and Section Formula

Distance Between Two Points

\[ P(x_1,y_1,z_1),\quad Q(x_2,y_2,z_2) \] \[ PQ= \sqrt{ (x_2-x_1)^2+ (y_2-y_1)^2+ (z_2-z_1)^2 } \]

Section Formula

If \(P\) divides \(A(x_1,y_1,z_1)\) and \(B(x_2,y_2,z_2)\) internally in the ratio \(m:n\), then

\[ P= \left( \frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}, \frac{mz_2+nz_1}{m+n} \right) \]

Direction

Direction Cosines

If a directed line makes angles \(\alpha,\beta,\gamma\) with the positive \(x,y,z\)-axes respectively, then

\[ l=\cos\alpha,\qquad m=\cos\beta,\qquad n=\cos\gamma \] \[ \boxed{l^2+m^2+n^2=1} \]

\(l\)

Direction cosine with x-axis.

\(m\)

Direction cosine with y-axis.

\(n\)

Direction cosine with z-axis.

Direction Ratios

Any three numbers proportional to the direction cosines of a line are called its direction ratios.

\[ a:b:c=l:m:n \] If \(a,b,c\) are direction ratios, then the corresponding direction cosines are \[ l=\frac{a}{\sqrt{a^2+b^2+c^2}}, \quad m=\frac{b}{\sqrt{a^2+b^2+c^2}}, \quad n=\frac{c}{\sqrt{a^2+b^2+c^2}} \]

Key idea: Direction ratios need not be unique. Any non-zero scalar multiple of \((a,b,c)\) gives the same direction.

Equation of a Line

Equation of a Line in Space

Vector Form

\[ \boxed{ \vec r=\vec a+\lambda\vec b } \]

Here \(\vec a\) is the position vector of a point on the line and \(\vec b\) is a vector parallel to the line.

Cartesian Form

\[ \boxed{ \frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} } \]

Lines in Space

Angle Between Two Lines

If two lines have direction ratios \((a_1,b_1,c_1)\) and \((a_2,b_2,c_2)\), then the acute angle \(\theta\) between them satisfies

\[ \boxed{ \cos\theta= \frac{ |a_1a_2+b_1b_2+c_1c_2| }{ \sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2} } } \]

Parallel Lines

\[ a_1:b_1:c_1 = a_2:b_2:c_2 \]

Perpendicular Lines

\[ a_1a_2+b_1b_2+c_1c_2=0 \]

Distance in Space

Skew Lines and Shortest Distance

Two lines which are neither parallel nor intersecting are called skew lines. The shortest distance between skew lines is measured along their common perpendicular.

For \[ L_1:\vec r=\vec a_1+\lambda\vec b_1 \] \[ L_2:\vec r=\vec a_2+\mu\vec b_2 \] the shortest distance is \[ \boxed{ d= \frac{ |(\vec a_2-\vec a_1) \cdot (\vec b_1\times\vec b_2)| }{ |\vec b_1\times\vec b_2| } } \]

Plane Geometry

Equation of a Plane

Vector Form

\[ \boxed{ (\vec r-\vec a)\cdot\vec n=0 } \]

Cartesian Form

\[ \boxed{ ax+by+cz+d=0 } \]

Plane Through a Point

A plane passing through \(P(x_1,y_1,z_1)\) and having normal vector \(a\hat i+b\hat j+c\hat k\) is

\[ \boxed{ a(x-x_1)+b(y-y_1)+c(z-z_1)=0 } \]

Planes

Angle Between Two Planes

If the equations of two planes are

\[ a_1x+b_1y+c_1z+d_1=0 \] \[ a_2x+b_2y+c_2z+d_2=0 \]

then the angle \(\theta\) between the planes is

\[ \boxed{ \cos\theta= \frac{ |a_1a_2+b_1b_2+c_1c_2| }{ \sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2} } } \]

Parallel Planes

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

Perpendicular Planes

\[ a_1a_2+b_1b_2+c_1c_2=0 \]

Distance

Distance of a Point from a Plane

For the point \(P(x_1,y_1,z_1)\) and plane

\[ ax+by+cz+d=0 \]

the perpendicular distance is

\[ \boxed{ D= \frac{ |ax_1+by_1+cz_1+d| }{ \sqrt{a^2+b^2+c^2} } } \]

Important: The denominator is the magnitude of the normal vector to the plane.

Worked Problems

Solved Examples

Example 1 — Direction Cosines

Find the direction cosines of a line whose direction ratios are \(2,-3,6\).

Example 2 — Equation of a Line

Find the equation of the line passing through \(A(2,-1,3)\) and having direction ratios \(4,2,-5\).

Example 3 — Point to Plane Distance

Find the distance of \(P(1,-2,3)\) from the plane \[ 2x-y+2z-5=0. \]

Example 4 — Angle Between Lines

Find the angle between lines having direction ratios \((1,2,2)\) and \((2,-1,2)\).

Higher Order Thinking

HOT Questions

HOT 1

A line has direction cosines \(l,m,n\). If \(l=m\), determine the possible relationship between \(l\) and \(n\) when the line makes an angle of \(60^\circ\) with the z-axis.

HOT 2

A plane passes through the point \((1,2,-1)\) and is perpendicular to the line whose direction ratios are \(2,-1,3\). Determine its equation.

HOT 3

Determine whether the lines \[ \frac{x-1}{2} = \frac{y+2}{3} = \frac{z-4}{-1} \] and \[ \frac{x-3}{4} = \frac{y-1}{6} = \frac{z+2}{-2} \] are parallel.

HOT 4

Find the condition on \(a,b,c\) so that the line with direction ratios \(a,b,c\) is perpendicular to the line with direction ratios \(2,-3,4\).

HOT 5

Explain why the equation \[ ax+by+cz+d=0 \] represents a plane and identify a normal vector to the plane.

Last Minute Revision

Quick Revision

Direction

• \(l^2+m^2+n^2=1\)
• \(l,m,n\) are direction cosines.
• \(a,b,c\) are direction ratios.
• \(l:m:n=a:b:c\)

Lines

• \(\vec r=\vec a+\lambda\vec b\)
• Cartesian line form.
• Use dot product for perpendicularity.
• Use direction ratios for angles.

Planes

• \(ax+by+cz+d=0\)
• Normal vector \(=(a,b,c)\)
• Plane angle is angle between normals.
• Distance uses perpendicular formula.

Distances

• Point-to-point distance.
• Point-to-plane distance.
• Shortest distance between skew lines.
• Always identify the normal/common perpendicular direction.

Formula Bank

Formula Sheet

Distance

\[ PQ= \sqrt{ (x_2-x_1)^2+ (y_2-y_1)^2+ (z_2-z_1)^2 } \]

Direction Cosines

\[ l^2+m^2+n^2=1 \]

Line

\[ \vec r=\vec a+\lambda\vec b \] \[ \frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} \]

Plane

\[ ax+by+cz+d=0 \]

Point-Plane Distance

\[ D= \frac{|ax_1+by_1+cz_1+d|} {\sqrt{a^2+b^2+c^2}} \]

Shortest Distance

\[ d= \frac{ |(\vec a_2-\vec a_1) \cdot (\vec b_1\times\vec b_2)| }{ |\vec b_1\times\vec b_2| } \]

Exam Strategy

Exam Tips

01. Identify the Object

First decide whether the question concerns a point, line or plane.

02. Identify Direction

Extract direction ratios before applying angle or perpendicularity conditions.

03. For a Plane Find the Normal

In \(ax+by+cz+d=0\), the normal vector is \((a,b,c)\).

04. Check the Geometry

Parallel, perpendicular, intersecting and skew conditions should be checked before selecting a distance formula.

05. Avoid Sign Errors

Carefully preserve the signs of direction ratios and coordinates.

06. Verify

For direction cosines verify \(l^2+m^2+n^2=1\).

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