The projections of a line segment on $x, y, z$ axes are $12, 4, 3$ . The length and the direction cosines of the line segments are

13, < 12 / 13, 4 / 13, 3 / 13 >

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19, < 12 / 19, 4 / 19, 19 >

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11, < 12 / 11, 4 / 11, 3 / 11 >

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None of these

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Solution

The correct option is A $13, < 12 / 13, 4 / 13, 3 / 13 >$
Let the line segment be represented in vector form as $\to a = a_{1} \to i + a_{2} \to j + a_{3} \to k$
Vector along coordinate axes are $\to i, \to j, \to k$
Vector along Given that projection of $\to a$ on x axis is 12, that with y axis is 4 and that with z axis is 3.
$⟹ \to a . \to i = 12 ⟹ a_{1} = 12.$
Similarily $a_{2} = 4, a_{3} = 3$
$∴ \to a = 12 \to i + 4 \to j + 3 \to k$
The length of the line segment $= | \to a | = \sqrt{144 + 16 + 9} = 13$
and the direction cosines of the line segments are
$< \frac{12}{13}, \frac{4}{13}, \frac{3}{13} >$

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13; \frac{12}{13}, \frac{4}{13}, \frac{3}{13}

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