A line makes a 45^{circ} angle with the posit | vedclass

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Question

(C) Let the angles made by the line with the positive $X$,$Y$,and $Z$ axes be $\alpha$,$\beta$,and $\gamma$ respectively. Given $\alpha = 45^{\circ}$

Vedclass · Accepted Answer

$165$

Vedclass · Answer

(C) Let the angles made by the line with the positive $X$,$Y$,and $Z$ axes be $\alpha$,$\beta$,and $\gamma$ respectively. Given $\alpha = 45^{\circ}$ and $\beta = \gamma$. Using the direction cosine property $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$: $\cos^2 45^{\circ} + \cos^2 \beta + \cos^2 \beta = 1$ $\left(\frac{1}{\sqrt{2}}\right)^2 + 2\cos^2 \beta = 1$ $\frac{1}{2} + 2\cos^2 \beta = 1$ $2\cos^2 \beta = \frac{1}{2}$ $\cos^2 \beta = \frac{1}{4}$ $\cos \beta = \frac{1}{2}$ (since angles are acute) $\beta = 60^{\circ}$. Thus,$\beta = 60^{\circ}$ and $\gamma = 60^{\circ}$. The sum of the angles is $\alpha + \beta + \gamma = 45^{\circ} + 60^{\circ} + 60^{\circ} = 165^{\circ}$.

$A$ line makes a $45^{\circ}$ angle with the positive $X$-axis and makes equal angles with the positive $Y$-axis and $Z$-axis respectively. The sum of the three angles which the line makes with the positive $X$-axis,$Y$-axis,and $Z$-axis is: (in $^{\circ}$)

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