3D Geometry
Direction Cosines and Angles
Grade 12
Question:
<p>If the direction cosines of two lines are such that <span class="math">l + m + n = 0</span> and <span class="math">l^2 + m^2 - n^2 = 0</span>, then the angle between them is</p>
<p>(a) <span class="math">\pi</span></p>
<p>(b) <span class="math">\frac{\pi}{3}</span></p>
<p>(c) <span class="math">\frac{\pi}{4}</span></p>
<p>(d) <span class="math">\frac{\pi}{6}</span></p>
Step-by-Step Solution
Key Concept: Use the given constraints on direction cosines to find the relationship between l, m, n, then calculate the angle using the direction cosine formula.
Step 1: From the condition l + m + n = 0 , we get n = -(l + m) Step 2: Substitute into l^2 + m^2 - n^2 = 0 : l^2 + m^2 - (l + m)^2 = 0 Step 3: Expanding: l^2 + m^2 - l^2 - 2lm - m^2 = 0 , which gives -2lm = 0 Step 4: This means either l = 0 or m = 0 . This allows us to find specific direction cosines for the two lines. Step 5: The angle between the two lines can be found using the dot product formula: \cos \theta = \frac{\pi}{3} ∴ Answer is (b).
Correct Answer: B