If a line makes angles $α,β$ and $γ$ with the positive directions of x-axis, y-axis and z-axis respectively, then $\sin^2α + \sin^2 β+ \sin^2 γ$ is equal to

2

The correct answer is Option (3) → 2

Given: A line makes angles $\alpha, \beta, \gamma$ with the positive directions of the x-, y-, and z-axes respectively.

Let the direction cosines of the line be $l = \cos \alpha$, $m = \cos \beta$, and $n = \cos \gamma$.

We know from the identity of direction cosines:

$l^2 + m^2 + n^2 = 1$

Now we are asked to find: $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma$

Using the identity $\sin^2 \theta = 1 - \cos^2 \theta$, we get:

$\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 3 - (\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma)$

$= 3 - (l^2 + m^2 + n^2) = 3 - 1 = 2$