Introduction to 3-D Geometry

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Question

A line makes an angle of $60 °$ with each of x and y axis, the which it makes with $z$ axis is ( in degrees)

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Solution

Let $α$ be the angle which the line makes with $z$ -axis, thus the direction cosines of the line are $\cos 60^{\circ}, \cos 60^{\circ}, \cos α$
$\begin{array}{l} \Rightarrow \cos^{2} 60^{\circ} + \cos^{2} 60^{\circ} + \cos^{2} α = 1 \\ \Rightarrow \cos^{2} α = 1 - 1 / 4 - 1 / 4 = 1 / 2 \end{array}$
$\Rightarrow \cos α = \pm 1 / \sqrt{2}$ so $α = 45^{\circ} .$

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Introduction to 3-D Geometry

Remember concepts with our Masterclasses.

A line makes an angle of 60° with each of x and y axis, the which it makes with z axis is ( in degrees)

Let α be the angle which the line makes with z-axis, thus the direction cosines of the line are cos ⁡60∘,cos ⁡60∘,cos ⁡α⇒ cos2⁡60∘+cos2⁡60∘+cos2⁡α=1⇒ cos2⁡α=1−1/4−1/4=1/2⇒ cos⁡α=±1/2so α=45∘.

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A line makes an angle of $60 °$ with each of x and y axis, the which it makes with $z$ axis is ( in degrees)