If l, m, n are the direction cosines of a straight line, then prove that $l^{2} + m^{2} + n^{2} = 1$ .

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Solution

Let $O P$ be any line through the origin $O$ which has direction cosines $l, m, n$ .

Let $P$ be the point having coordinates $(x, y, z)$ and $O P = r$ .

Then $O P^{2} = x^{2} + y^{2} + z^{2} = r^{2} . . . (1)$

From $P$ draw $P A, P B, P C$ perpendicular on the coordinate axes, so that $O A = x, O B = y, O C = z$

Also, $∠ P O A = α, ∠ P O B = β, ∠ P O C = γ$

From triangle $A O P, l = c o s α = \frac{x}{r} \Rightarrow x = l r$

Similarly, $y = m r, z = n r$

Adding all we get $x^{2} + y^{2} + z^{2} = r^{2} (l^{2} + m^{2} + n^{2})$

$\Rightarrow r^{2} = r^{2} (l^{2} + m^{2} + n^{2})$

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

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If l, m, n are the direction cosines of a straight line, then prove that l2+m2+n2=1.

If l, m, n are the direction cosines of a straight line, then prove that $l^{2} + m^{2} + n^{2} = 1$ .