Question

Find the direction cosines of a line which makes equal angles with the coordinate axes

Answer 1

Answer:

$± \frac{1}{ \sqrt{3} }, \: \: ±\frac{1}{ \sqrt{3} }, \: \: ± \frac{1}{ \sqrt{3} }$

Step-by-step explanation:

To find the direction cosines of a line which makes equal angles with the coordinate axes;

We know that Direction cosine of a line are

$l, \: \: m, \: \: n \:$

given by

$l = \cos( \alpha ) \\ \\ m = \cos( \beta ) \\ \\ n = \cos( \gamma )$

here

$\alpha \: \: \beta \: \: \: \gamma \: \:$

are the angles made with x,y and z axis respectively.

let the line make

$\alpha$

angle with all three axis,

we also know that

${l}^{2} + {m}^{2} + {n}^{2} = 1 \\ \\ so \\ \\ {cos}^{2} \alpha + {cos}^{2} \alpha + {cos}^{2} \alpha = 1 \\ \\ 3{cos}^{2} \alpha = 1 \\ \\ {cos}^{2} \alpha = \frac{1}{3} \\ \\ cos \: \alpha = ±\frac{1}{ \sqrt{3} }$

So,direction cosine of that line would be

$±\frac{1}{ \sqrt{3} } , \: \: ±\frac{1}{ \sqrt{3} }, \: \: ±\frac{1}{ \sqrt{3} }$

Hope it helps you.

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

TO DETERMINE

The direction cosines of a line which makes equal angles with the coordinate axes

CALCULATION

Let a line AB makes equal angles with the coordinate axes

$\sf{Let \: \: \alpha \: \: be \: the \: angle}$

$\sf{ Let \: \: l \: , \: m \: , \: n \: \: be \: the \: direction \: cosines} \:$

$\sf{ Then \: \: \: l \: = m = n = \cos \alpha \: }$

We know that

$\sf{ {l}^{2} + {m}^{2} + {n}^{2} = 1\: \: }$

$\implies \: \sf{ { \cos}^{2} \alpha + \:{ \cos}^{2} \alpha + { \cos}^{2} \alpha = 1\: \: }$

$\implies \: \sf{ 3{ \cos}^{2} \alpha = 1\: \: }$

$\displaystyle \: \implies \: \sf{ { \cos}^{2} \alpha = \frac{1}{3} \: \: }$

$\displaystyle \: \implies \: \sf{ { \cos} \alpha = \pm \: \frac{1}{ \sqrt{3} } \: \: }$

RESULT

Hence The direction cosines of a line which makes equal angles with the coordinate axes are

$\displaystyle \: \sf{ \frac{1}{ \sqrt{3} } \: , \frac{1}{ \sqrt{3}} \: , \frac{1}{ \sqrt{3} } } \: \: \: or \: \: \: - \frac{1}{ \sqrt{3} } \: , - \frac{1}{ \sqrt{3}} \: , - \frac{1}{ \sqrt{3} }$

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