CBSE Sample Paper 2020-21 · Part A: Section I · 1 mark

Q13: Direction cosines of the normal to the YZ plane

Sample 2020-21Ch 11 · Three Dimensional Geometry

The question

Find the direction cosines of the normal to YZ plane?

Exam answer

The YZ plane is $x = 0$ ⁠; its normal is along the x-axis.

Why?

The YZ plane contains the y-axis and the z-axis, and every point on it has $x = 0$ ⁠.

A normal is perpendicular to every line in the plane. The x-axis is perpendicular to both the y-axis and the z-axis: $\hat i\cdot\hat j = 0$ and $\hat i\cdot\hat k = 0$ ⁠.

The x-axis makes angles $0, \frac{\pi}{2}, \frac{\pi}{2}$ with the x, y and z axes.

\begin{aligned} l &= \cos 0 = 1 \\ m &= \cos\frac{\pi}{2} = 0 \\ n &= \cos\frac{\pi}{2} = 0 \end{aligned}

Why?

The direction cosines of a line are the cosines of the angles $\alpha, \beta, \gamma$ it makes with the x, y and z axes:

l = \cos\alpha,\ m = \cos\beta,\ n = \cos\gamma

Check: $l^2 + m^2 + n^2 = 1 + 0 + 0 = 1$ ⁠, as it must be.

$\therefore$ the direction cosines are $1, 0, 0$ ⁠.

Where this comes from

Chapter 11, Three Dimensional Geometry: Direction cosines of the normal to the YZ-plane.

Question type: Direction cosines from angles or from ratios

How to spot it: click to see

Type 1. Direction cosines from angles or from ratios

Spot it: you are given the angles with the axes, or three direction ratios, and asked for the direction cosines.

Here: The normal to the YZ plane is the x-axis, which makes angles $0, \frac{\pi}{2}, \frac{\pi}{2}$ with the axes. Take the cosine of each: $1, 0, 0$ ⁠.

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Three Dimensional Geometry in past papers: question 32 of 35 · see them all