If a line makes angles Q1, Q21 and Q3 respectively with the coordinate axis then the value of cos² Q1 + cos² Q2 + cos² Q3

The correct option is 1.

Let us understand why this is the correct answer step-by-step.

In three-dimensional coordinate geometry, if a line makes angles $Q_1$, $Q_2$, and $Q_3$ (often denoted as $\alpha$, $\beta$, and $\gamma$) with the positive directions of the x-axis, y-axis, and z-axis respectively, these angles are known as the direction angles of the line.

The cosines of these angles, namely $\cos Q_1$, $\cos Q_2$, and $\cos Q_3$, are called the direction cosines of the line. They are commonly represented by the letters $l$, $m$, and $n$ respectively:
$l=\cos Q_1$
$m=\cos Q_2$
$n=\cos Q_3$

Let $\overrightarrow{u}$ be a unit vector along the given line. We can express this unit vector in terms of its components along the coordinate axes using its direction cosines as follows:
$\overrightarrow{u}=l\hat{i}+m\hat{j}+n\hat{k}$

Since $\overrightarrow{u}$ is a unit vector, its magnitude must be equal to 1. The magnitude of a vector is given by the square root of the sum of the squares of its components:
$\left|\overrightarrow{u}\right|=\sqrt{l^2+m^2+n^2}=1$

Squaring both sides of this equation, we get the fundamental identity for direction cosines:
$l^2+m^2+n^2=1$

Substituting the values of $l$, $m$, and $n$ back into this equation, we get:
${\cos }^2{Q}_1+{\cos }^2{Q}_2+{\cos }^2{Q}_3=1$

Thus, the value of ${\cos }^2{Q}_1+{\cos }^2{Q}_2+{\cos }^2{Q}_3$ is always equal to 1.