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Class 12 Maths Notes

Three Dimensional Geometry Class 12 Notes

Complete, exam-ready notes on three dimensional geometry: direction cosines, equations of lines and planes, angle and distance between them — written for CBSE boards and JEE revision.

Class12SubjectMathematicsCoversCBSE · JEE

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

What is this chapter about in one line?

Three dimensional geometry writes lines and planes as vector or cartesian equations, then measures the angles and distances between them.

Direction Cosines and Direction Ratios

Direction cosines

The direction cosines l, m, n of a line are the cosines of its angles with the x, y, z axes. They satisfy l2+m2+n2=1l^2 + m^2 + n^2 = 1. Any set of numbers proportional to (l, m, n) is a set of direction ratios.

l=cos⁡α,m=cos⁡β,n=cos⁡γ,l2+m2+n2=1l = \cos\alpha,\quad m = \cos\beta,\quad n = \cos\gamma,\qquad l^2 + m^2 + n^2 = 1
Direction cosines

Equation of a Line

Cartesian equation

A line through (x1,y1,z1)(x_1,y_1,z_1) with direction ratios a, b, c is x−x1a=y−y1b=z−z1c\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}; if a direction ratio is zero, that numerator is set to zero instead.

r⃗=a⃗+tb⃗\vec{r} = \vec{a} + t\vec{b}
Vector equation of a line

Equation of a Plane

Plane

A plane perpendicular to the normal vector (a,b,c)(a,b,c) through a point is a(x−x0)+b(y−y0)+c(z−z0)=0a(x-x_0) + b(y-y_0) + c(z-z_0) = 0, written compactly as ax+by+cz=dax + by + cz = d.

xp+yq+zr=1,d=∣ax0+by0+cz0−d′∣a2+b2+c2\frac{x}{p} + \frac{y}{q} + \frac{z}{r} = 1,\qquad d = \frac{|ax_0 + by_0 + cz_0 - d'|}{\sqrt{a^2 + b^2 + c^2}}
Intercept form and perpendicular distance

Angles and Distances

  • Angle between two lines follows from their direction ratios via the dot-product formula.
  • Angle between two planes is the angle between their normals.
  • Angle between a line and a plane is the complement of the angle between the line and the plane's normal.
  • Two lines are skew when they are neither parallel nor intersecting; their shortest distance uses a common perpendicular.

Solved Examples

Example: Find the direction cosines of the line with direction ratios 3, 4, 12.

Solution: Magnitude =9+16+144=13= \sqrt{9+16+144} = 13 (note 3-4-12-13), so the direction cosines are 3/13, 4/13, 12/13, and (3/13)2+(4/13)2+(12/13)2=1(3/13)^2 + (4/13)^2 + (12/13)^2 = 1.

Example: Find the perpendicular distance of (1,1,1)(1,1,1) from the plane 2x−y+2z=62x - y + 2z = 6.

Solution: d=∣2(1)−1+2(1)−6∣4+1+4=∣−3∣3=1d = \frac{|2(1) - 1 + 2(1) - 6|}{\sqrt{4+1+4}} = \frac{|-3|}{3} = 1 unit.

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

Direction cosines identity

l2+m2+n2=1l^2 + m^2 + n^2 = 1

Line through a point

x−x1a=y−y1b=z−z1c\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}

Plane with normal

a(x−x0)+b(y−y0)+c(z−z0)=0a(x-x_0) + b(y-y_0) + c(z-z_0) = 0

Perpendicular distance

∣ax1+by1+cz1−d∣a2+b2+c2\frac{|ax_1 + by_1 + cz_1 - d|}{\sqrt{a^2+b^2+c^2}}

Angle between lines

cos⁡θ=a1a2+b1b2+c1c2a12+b12+c12a22+b22+c22\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}}

Angle between planes

cos⁡θ=a1a2+b1b2+c1c2a12+b12+c12a22+b22+c22\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}}

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

  • Always check l² + m² + n² = 1 for direction cosines — it is the consistency check.
  • Parallel lines and planes have proportional direction ratios / normals.
  • Perpendicular distance needs the plane in the form ax + by + cz = d.
  • Angle of a line with a plane = 90° − angle of the line with the plane's normal.
  • Skew lines never meet and are not parallel; their shortest distance comes from a common perpendicular.

FAQ

Common questions

What is the difference between direction cosines and direction ratios?

Direction cosines are the exact cosines with the axes and satisfy l² + m² + n² = 1. Direction ratios are any numbers proportional to them, usually the components of the direction vector.

How do I find the equation of a line in 3D?

Use a point on the line and its direction ratios: (x − x₁)/a = (y − y₁)/b = (z − z₁)/c, or in vector form r = a + tb.

How do you find the distance of a point from a plane?

Put the plane in the form ax + by + cz = d, substitute the point, and take the absolute value over the normal's magnitude: |ax₁ + by₁ + cz₁ − d|/√(a² + b² + c²).

What are skew lines?

Lines in three dimensions that are neither parallel nor intersecting — they lie in different directions and never touch, so they have a shortest connecting perpendicular.

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