If O equiv (0, 0, 0) and P equiv (1, sqrt{2}, | vedclass

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(D) The direction ratios of the line $OP$ are $(1, \sqrt{2}, 1)$.Let the direction cosines of the line $OP$ be $(l, m, n)$.The magnitude of vector $\v

Vedclass · Accepted Answer

$30^{\circ}, 30^{\circ}, 45^{\circ}$

Vedclass · Answer

(D) The direction ratios of the line $OP$ are $(1, \sqrt{2}, 1)$.Let the direction cosines of the line $OP$ be $(l, m, n)$.The magnitude of vector $\vec{OP}$ is $\sqrt{1^2 + (\sqrt{2})^2 + 1^2} = \sqrt{1 + 2 + 1} = \sqrt{4} = 2$.Thus,the direction cosines are $l = \frac{1}{2}$,$m = \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}$,and $n = \frac{1}{2}$.The angle $\alpha$ made by a line with direction cosines $(l, m, n)$ with the $YOZ$ plane is given by $\sin \alpha = |l| = \frac{1}{2}$,so $\alpha = 30^{\circ}$.The angle $\beta$ made with the $ZOX$ plane is given by $\sin \beta = |m| = \frac{1}{\sqrt{2}}$,so $\beta = 45^{\circ}$.The angle $\gamma$ made with the $XOY$ plane is given by $\sin \gamma = |n| = \frac{1}{2}$,so $\gamma = 30^{\circ}$.Wait,re-evaluating the question order: $XOY$ plane (involves $z$),$YOZ$ plane (involves $x$),$ZOX$ plane (involves $y$).Angle with $XOY$ plane: $\sin 	heta_1 = |n| = \frac{1}{2} \implies 	heta_1 = 30^{\circ}$.Angle with $YOZ$ plane: $\sin 	heta_2 = |l| = \frac{1}{2} \implies 	heta_2 = 30^{\circ}$.Angle with $ZOX$ plane: $\sin 	heta_3 = |m| = \frac{1}{\sqrt{2}} \implies 	heta_3 = 45^{\circ}$.Given the options,there might be a convention difference. Let's re-calculate: The angle $	heta$ between a line with direction cosines $(l, m, n)$ and a plane with normal $(a, b, c)$ is $\sin 	heta = |al + bm + cn|$.For $XOY$ ($z=0$,normal $(0,0,1)$): $\sin 	heta_1 = |0(1/2) + 0(1/\sqrt{2}) + 1(1/2)| = 1/2 \implies 30^{\circ}$.For $YOZ$ ($x=0$,normal $(1,0,0)$): $\sin 	heta_2 = |1(1/2) + 0 + 0| = 1/2 \implies 30^{\circ}$.For $ZOX$ ($y=0$,normal $(0,1,0)$): $\sin 	heta_3 = |0 + 1(1/\sqrt{2}) + 0| = 1/\sqrt{2} \implies 45^{\circ}$.Correct answer is $30^{\circ}, 30^{\circ}, 45^{\circ}$.

If $O \equiv (0, 0, 0)$ and $P \equiv (1, \sqrt{2}, 1)$,then the acute angles made by the line $OP$ with $XOY$,$YOZ$,and $ZOX$ planes are,respectively:

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