3D Geometry
Direction Cosines and Direction Ratios
Grade 12
Question:
<p>If a line makes angles α, β and γ with the coordinate axes, then:</p>
<p>(a) \(\cos 2\alpha + \cos 2\beta + \cos 2\gamma - 1 = 0\)</p>
<p>(b) \(\cos 2\alpha + \cos 2\beta + \cos 2\gamma - 2 = 0\)</p>
<p>(c) \(\cos 2\alpha + \cos 2\beta + \cos 2\gamma + 1 = 0\)</p>
<p>(d) \(\cos 2\alpha + \cos 2\beta + \cos 2\gamma + 2 = 0\)</p>
Step-by-Step Solution
Key Concept: Use the property that direction cosines satisfy \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\) and convert using the double angle formula \(\cos 2\theta = 2\cos^2 \theta - 1\).
Solution: If \(\cos \alpha, \cos \beta\) and \(\cos \gamma\) are the direction cosines of a line, then: \[\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\] This can be rewritten as: \[1 + \cos 2\alpha + 1 + \cos 2\beta + 1 + \cos 2\gamma = 2\] \[\cos 2\alpha + \cos 2\beta + \cos 2\gamma + 1 = 0\] ∴ Answer is (c).
Correct Answer: C