Maths

Class 12

Maths · Class 12 · Chapter 11

Three Dimensional Geometry

9 lessons 72 min 2 simulations

This chapter puts the vectors of the previous one to work on lines in space. A line's direction is carried by three numbers, its direction cosines or any direction ratios, and a line is fixed by one point and a direction: r = a + λb, or (x − x₁)/a = (y − y₁)/b = (z − z₁)/c. From there the chapter finds the angle between two lines from the dot product of their directions, and the shortest distance between two lines, which in space may be skew: neither parallel nor meeting. The cross product of the two directions gives the one direction perpendicular to both, and the gap is the projection of any joining segment on it. The rationalised text stops at lines; the last two sections add the foot of a perpendicular and the plane, which JEE still asks.

Start

What the exam asks

JEE Main asks for direction cosines from ratios or from two points, collinearity through proportional ratios, a line's vector and Cartesian forms and the switch between them, the angle between two lines, the perpendicular and parallel conditions, and the shortest distance between skew and parallel lines. Foot of the perpendicular, image of a point in a line, and planes (normal form, angle, distance of a point) are outside the rationalised text but still asked in JEE; they are the last two sections. Marks go on the ± sign of the direction cosines, on taking |cos θ| for the acute angle, on the modulus in the shortest-distance formula, and on reading a line whose Cartesian form hides a coefficient, like (1 − x)/3.

Lessons

9 lessons · 72 min