A line makes angles frac{alpha}{2}, frac{beta | vedclass

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(D) Let the direction cosines of the line be $l, m, n$. Since the line makes angles $\frac{\alpha}{2}, \frac{\beta}{2}, \frac{\gamma}{2}$ with the axe

Vedclass · Accepted Answer

$-1$

Vedclass · Answer

(D) Let the direction cosines of the line be $l, m, n$. Since the line makes angles $\frac{\alpha}{2}, \frac{\beta}{2}, \frac{\gamma}{2}$ with the axes,we have $l = \cos(\frac{\alpha}{2}), m = \cos(\frac{\beta}{2}), n = \cos(\frac{\gamma}{2})$.Since $l^2 + m^2 + n^2 = 1$,we have $\cos^2(\frac{\alpha}{2}) + \cos^2(\frac{\beta}{2}) + \cos^2(\frac{\gamma}{2}) = 1$.We know that $\cos 	heta = 2 \cos^2(\frac{	heta}{2}) - 1$,so $\cos^2(\frac{	heta}{2}) = \frac{1 + \cos 	heta}{2}$.Substituting this into the identity:$\frac{1 + \cos \alpha}{2} + \frac{1 + \cos \beta}{2} + \frac{1 + \cos \gamma}{2} = 1$$\frac{3 + \cos \alpha + \cos \beta + \cos \gamma}{2} = 1$$3 + \cos \alpha + \cos \beta + \cos \gamma = 2$$\cos \alpha + \cos \beta + \cos \gamma = 2 - 3 = -1$.

$A$ line makes angles $\frac{\alpha}{2}, \frac{\beta}{2}, \frac{\gamma}{2}$ with the positive directions of the coordinate axes ($x, y, z$ axes respectively). Then,the value of $\cos \alpha + \cos \beta + \cos \gamma$ is:

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