Introduction to 3-D Geometry

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Question

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

Moderate

Solution

$\begin{array}{l} \cos^{2} 45^{\circ} + \cos^{2} 120^{\circ} + \cos^{2} θ = 1 \\ \Rightarrow \cos^{2} θ = 1 - (1 / 2) - (- 1 / 2)^{2} = 1 / 4 \\ \Rightarrow \cos θ = 1 / 2 \Rightarrow θ = 60^{\circ} \end{array}$

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