CBSE Class 12 Maths

Three Dimensional Geometry

Each concept is a short lesson, then questions until you get three right.

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6 concepts, a short lesson each

Dot product0/1
Lines0/5

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Direction cosines and direction ratios

$l = \cos\alpha$ , $m = \cos\beta$ , $n = \cos\gamma$ with $l^2 + m^2 + n^2 = 1$ ; ratios $a : b : c$ scale to cosines by $\dfrac{1}{\sqrt{a^2 + b^2 + c^2}}$

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Dot product0 of 1

Direction cosines and direction ratios $l = \cos\alpha$ , $m = \cos\beta$ , $n = \cos\gamma$ with $l^2 + m^2 + n^2 = 1$ ; ratios $a : b : c$ scale to cosines by $\dfrac{1}{\sqrt{a^2 + b^2 + c^2}}$ Start

Lines0 of 5

Vector and Cartesian equations of a lineThrough $\vec a$ along $\vec b$ : $\vec r = \vec a + \lambda\vec b$ , or $\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}$ ; through two points, take $\vec b = \vec B - \vec A$ Start
Angle between two lines $\cos\theta = \dfrac{|\vec b_1\cdot\vec b_2|}{|\vec b_1||\vec b_2|}$ from the direction vectors; perpendicular when $a_1a_2 + b_1b_2 + c_1c_2 = 0$ Start
Distance of a point from a lineWrite the foot as $\vec a + \lambda\vec b$ , make $\overrightarrow{PM}\cdot\vec b = 0$ to find $\lambda$ , then measure $PM$ Start
Shortest distance between two linesSkew lines: $d = \dfrac{|(\vec a_2 - \vec a_1)\cdot(\vec b_1\times\vec b_2)|}{|\vec b_1\times\vec b_2|}$ ; parallel lines: $d = \dfrac{|(\vec a_2 - \vec a_1)\times\vec b|}{|\vec b|}$ Start
Intersecting and coplanar linesTwo lines are coplanar (meet or are parallel) when $(\vec a_2 - \vec a_1)\cdot(\vec b_1\times\vec b_2) = 0$ ; find the meeting point by equating the parametersStart

Practise Three Dimensional Geometry

Once the concepts click, try the questions your exam asks.

Textbook exercisesEvery exercise, with model answers Past board questionsAsked from this chapter, with solutions