Three Dimensional Geometry Class 12: notes and important questions

Three Dimensional Geometry — revision notes

1. Direction cosines and ratios

• If a line makes angles \(\alpha, \beta, \gamma\) with the axes, \(l = \cos\alpha, m = \cos\beta, n = \cos\gamma\) and \(l^2 + m^2 + n^2 = 1\).
• Direction ratios \(a, b, c\) are proportional to \(l, m, n\): \(l = \dfrac{\pm a}{\sqrt{a^2 + b^2 + c^2}}\), etc. For \(PQ\): \(x_2 - x_1, y_2 - y_1, z_2 - z_1\).

2. Equation of a line

• Through \(\vec{a}\), parallel to \(\vec{b}\): \(\vec{r} = \vec{a} + \lambda\vec{b}\); Cartesian \(\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}\).
• Through two points: \(\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})\).
• Standard form needs coefficient \(1\) for \(x, y, z\): \(\dfrac{3 - x}{2} = \dfrac{x - 3}{-2}\), \(\dfrac{2z - 1}{6} = \dfrac{z - \frac12}{3}\).
• General point \((x_1 + a\lambda, y_1 + b\lambda, z_1 + c\lambda)\) — use it for points on the line, intersections and feet of perpendiculars.

3. Angle between two lines

\(\cos\theta = \dfrac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}}\). Perpendicular: \(a_1a_2 + b_1b_2 + c_1c_2 = 0\); parallel: \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\).

4. Shortest distance

• Skew lines (neither parallel nor intersecting): \(d = \dfrac{|(\vec{a}_2 - \vec{a}_1)\cdot(\vec{b}_1\times\vec{b}_2)|}{|\vec{b}_1\times\vec{b}_2|}\). \(d = 0\) (non-parallel) \(\iff\) the lines intersect.
• Parallel lines \(\vec{r} = \vec{a}_i + \lambda\vec{b}\): \(d = \dfrac{|\vec{b}\times(\vec{a}_2 - \vec{a}_1)|}{|\vec{b}|}\).
• Foot of perpendicular from \(P\) to a line: take the general point \(F\), solve \(\overrightarrow{PF}\cdot\vec{b} = 0\). Image \(P\prime = 2F - P\); distance \(= PF\).

Worked example 1

Line through \((1, 0, -1)\) and \((3, 2, 0)\): \(\dfrac{x - 1}{2} = \dfrac{y}{2} = \dfrac{z + 1}{1}\); direction cosines \(\tfrac23, \tfrac23, \tfrac13\).

Worked example 2

\(\vec{r} = \lambda\hat{k}\) and \(\vec{r} = \hat{i} + \mu\hat{j}\): \(\vec{b}_1\times\vec{b}_2 = \hat{k}\times\hat{j} = -\hat{i}\), \((\vec{a}_2 - \vec{a}_1)\cdot(-\hat{i}) = -1\), so \(d = 1\).

Worked example 3

Foot from \(P(0, 0, 3)\) to \(\vec{r} = \lambda(\hat{i} + \hat{j} + \hat{k})\): \((\lambda, \lambda, \lambda - 3)\cdot(1, 1, 1) = 3\lambda - 3 = 0\), foot \((1, 1, 1)\), distance \(\sqrt{1 + 1 + 4} = \sqrt6\).

Common errors

• Reading direction ratios before putting the line in standard form (signs and coefficients of \(x, y, z\)).
• Omitting the modulus in the angle formula (the angle between lines is taken acute) or in the distance formula.
• Using the skew-line formula for parallel lines (it gives \(0/0\)).
• Declaring lines intersecting after matching only two coordinates — always check the third.

Board-exam tips

• The 5-mark question is usually shortest distance, intersection, or foot/image of a point: learn the general-point method thoroughly.
• Planes (and angles/distances involving planes) are not in the current syllabus.