(A) Let the origin be $O(0, 0, 0)$ and the point be $P(2, 3, -1)$.The direction ratios of the line $OP$ are $(x_2 - x_1, y_2 - y_1, z_2 - z_1) = (2 -

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$\frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}}, \frac{-1}{\sqrt{14}}$

(A) Let the origin be $O(0, 0, 0)$ and the point be $P(2, 3, -1)$.The direction ratios of the line $OP$ are $(x_2 - x_1, y_2 - y_1, z_2 - z_1) = (2 - 0, 3 - 0, -1 - 0) = (2, 3, -1)$.The distance $OP$ is given by $\sqrt{2^2 + 3^2 + (-1)^2} = \sqrt{4 + 9 + 1} = \sqrt{14}$.The direction cosines $(l, m, n)$ are given by $\left(\frac{a}{\sqrt{a^2+b^2+c^2}}, \frac{b}{\sqrt{a^2+b^2+c^2}}, \frac{c}{\sqrt{a^2+b^2+c^2}}\right)$.Thus, the direction cosines are $\left(\frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}}, \frac{-1}{\sqrt{14}}\right)$.

Class 12 Mathematics Three Dimensional Geometry System of co-ordinates, Direction cosines and direction ratios, Projection

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The direction cosines of the line passing through $P(2, 3, -1)$ and the origin are

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A
$\frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}}, \frac{-1}{\sqrt{14}}$
B
$\frac{2}{\sqrt{14}}, \frac{-3}{\sqrt{14}}, \frac{1}{\sqrt{14}}$
C
$\frac{-2}{\sqrt{14}}, \frac{-3}{\sqrt{14}}, \frac{1}{\sqrt{14}}$
D
$\frac{-2}{\sqrt{14}}, \frac{3}{\sqrt{14}}, \frac{1}{\sqrt{14}}$

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