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Q.

A line AB in three dimensional space makes angles $45 °$ and $120 °$ with the positive x-axis and the positive y-axis respectively. If AB makes an acute angle $θ$ (in degrees) with the positive z-axis, then $θ$ equals

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a

$\frac{π}{4}$

b

$\frac{π}{2}$

c

$\frac{π}{3}$

d

$\frac{π}{6}$

answer is B.

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Detailed Solution

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We have,

$\begin{array}{l} l = \cos 45^{\circ} = \frac{1}{\sqrt{2}}, m = \cos 120^{\circ} = - \frac{1}{2} and n = \cos θ \\ ∴ l^{2} + m^{2} + n^{2} = 1 \\ \Rightarrow \frac{1}{2} + \frac{1}{4} + \cos^{2} θ = 1 \Rightarrow \cos^{2} θ = \frac{1}{4} \Rightarrow \cos θ = \frac{1}{2} [∵ θ is acute] \\ \Rightarrow θ = 60^{\circ} \end{array}$

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