The direction cosines of the line making angl | vedclass

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(A) Let the direction cosines be $l, m, n$. Given the angles $\alpha = \frac{\pi}{4}$, $\beta = \frac{\pi}{3}$, and $\gamma = 	heta$.Then $l = \cos \

Vedclass · Accepted Answer

$\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2}$

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(A) Let the direction cosines be $l, m, n$. Given the angles $\alpha = \frac{\pi}{4}$, $\beta = \frac{\pi}{3}$, and $\gamma = 	heta$.Then $l = \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}$ and $m = \cos \frac{\pi}{3} = \frac{1}{2}$.We know that $l^2 + m^2 + n^2 = 1$.Substituting the values, we get $\left(\frac{1}{\sqrt{2}}ight)^2 + \left(\frac{1}{2}ight)^2 + \cos^2 	heta = 1$.$\frac{1}{2} + \frac{1}{4} + \cos^2 	heta = 1$.$\frac{3}{4} + \cos^2 	heta = 1$.$\cos^2 	heta = 1 - \frac{3}{4} = \frac{1}{4}$.Since $0 Thus, the direction cosines are $\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2}$.

Column-$I$	Column-$II$
$A$. Tapetum	$i$. Degenerates during development
$B$. Intine	$ii$. Resistant organic material
$C$. Vegetative cell	$iii$. Pollen tube
$D$. Sporopollenin	$iv$. Provides nourishment to pollen grains

The direction cosines of the line making angles $\frac{\pi}{4}, \frac{\pi}{3}$ and $\theta$ $\left(0 < \theta < \frac{\pi}{2}\right)$ respectively with $x, y$ and $z$ axes, are

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