First slide

Direction ratios and direction cosines

Question

If a line makes an angle of $\frac{π}{4}$ with the positive direction of each of x-axis and y-axis, then the angle that the line makes with the positive direction of the z-axis is

Moderate

A

$\frac{π}{3}$
B

$\frac{π}{4}$
C

$\frac{π}{2}$
D

$\frac{π}{6}$

Solution

Here $l = \cos \frac{π}{4}, m = \cos \frac{π}{4}$
Let the line make an angle $γ$ with the z-axis. Then
$\begin{array}{l} l^{2} + m^{2} + n^{2} = 1 \\ or \cos^{2} \frac{π}{4} + \cos^{2} \frac{π}{4} + \cos^{2} γ = 1 \\ or \frac{1}{2} + \frac{1}{2} + \cos^{2} γ = 1 \\ or 2 \cos^{2} γ = 0 or \cos γ = 0 or γ = \frac{π}{2} \end{array}$

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