Question

If \( \alpha, \beta, \gamma \) are direction angles of a line and \( \alpha = 60^\circ, \beta = 45^\circ \), then \( \gamma \) is _________.

• A. \( 30^\circ \) or \( 90^\circ \)
• B. \( 45^\circ \) or \( 60^\circ \)
• C. \( 90^\circ \) or \( 130^\circ \)
• D. \( 60^\circ \) or \( 120^\circ \)

Hint:

Remember, the sum of the squares of the cosines of direction angles is always equal to 1:
\[ \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1 \]

Answer 1

The Correct Option is D

Step 1: Applying the Direction Cosine Identity
The identity for the sum of the squares of the direction cosines states: \[ \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1 \] Step 2: Substituting the Given Values
Substitute the known values into the equation: \[ \cos^2(60^\circ) + \cos^2(45^\circ) + \cos^2\gamma = 1 \] \[ \left( \frac{1}{4} \right) + \left( \frac{1}{2} \right) + \cos^2\gamma = 1 \] \[ \cos^2\gamma = \frac{1}{4} \] \[ \gamma = 60^\circ { or } 120^\circ \]