Question

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

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Answer 1

Let the direction cosines of the line make an angle α with each of the coordinate axes.

$\large\rm \therefore l = \cos \alpha, m = \cos \alpha, n = \cos \alpha$

$\large\rm$

we know that $\large\rm l^2 + m^2 + n^2 = 1$

$\large\rm \implies \cos^2 \alpha + \cos^2 \alpha + \cos^2 \alpha = 1$

$\large\rm \implies 3 \ \cos^2 \alpha = 1$

$\large\rm \implies \cos^2 \alpha = \dfrac{1}{3}$

$\large\rm \implies \cos \alpha = \pm \dfrac{1}{\sqrt{3}}$

$\large\rm$

Thus, the direction cosines of the line which is equally inclined to the coordinate axes are $\large\rm \pm \dfrac{1}{\sqrt{3}}, \pm \dfrac{1}{\sqrt{3}} , \pm \dfrac{1}{\sqrt{3}}$

Answer 2

Direction of cosines of a line making, $\alpha$ with x-axis, $\beta$ with y-axis, and $\gamma$ with z-axis are $\ell$,m,n respectively.

$\sf{\ell=cos\alpha,\:m=cos\beta,\:n=cos\gamma}$

Given, the line makes equal angles with co-ordinate axis.

So, $\alpha = \beta = \gamma$

Therefore, $\sf{\ell = m=n=cos\alpha}$

We know that,

$\sf{⇒\ell^2 + m² +n² = 1}$

$\sf{⇒cos²\alpha + cos²\alpha +cos²\alpha = 1}$

$\sf{⇒cos²\alpha = \dfrac{1}{3}}$

$\sf{⇒cos\alpha = ±\dfrac{1}{\sqrt{3}}}$

$\bold{∴\: \ell = ±\dfrac{1}{\sqrt{3}},\:m=±\dfrac{1}{\sqrt{3}},\:n=±\dfrac{1}{\sqrt{3}}}$