Question

If a line makes \(\alpha, \beta, \gamma\) with the positive direction of x, y and z-axes respectively. Then, \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma\) is equal to

• A. \(1/2\)
• B. \(-1/2\)
• C. \(-1\)
• D. 1

Hint:

Direction cosines: \(l = \cos\alpha, m = \cos\beta, n = \cos\gamma\), with \(l^2 + m^2 + n^2 = 1\).

Answer 1

The Correct Option is D

To solve the given problem, we need to understand the concept of direction cosines, which are associated with a line in 3D geometry. The direction cosines of a line, which makes angles \(\alpha\), \(\beta\), and \(\gamma\) with the positive directions of the x, y, and z-axes respectively, are given by \(\cos \alpha\), \(\cos \beta\), and \(\cos \gamma\).

According to the property of direction cosines, the sum of the squares of the direction cosines of a line is always equal to 1, i.e.,

\(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\)

Let's verify this by applying the known identity:

1. We consider the direction cosines as \(\cos \alpha\), \(\cos \beta\), and \(\cos \gamma\). These are derived from the unit vector in any random direction in 3D space.
2. By the definition of direction cosines, it follows that the sum of their squares is a fundamental identity, i.e.,
   • \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\)
3. Hence, based on the fundamental property of direction cosines, the value of \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma\) is indeed equal to 1.

Therefore, the correct answer is 1.