3D Geometry
Direction Cosines and Direction Ratios of a Line
Grade 12
Question:
<p>The two lines <em>x = ay + b, z = cy + d</em> and <em>x = a'y + b', z = c'y + d'</em> will be perpendicular, if and only if</p>
<p>\(aa' + bb' + cc' + 1 = 0\)</p>
<p>\(aa' + bb' + cc' = 0\)</p>
<p>\((a + a')(b + b') + (c + c') = 0\)</p>
<p>\(aa' + cc' + 1 = 0\)</p>
Step-by-Step Solution
Key Concept: Two lines in 3D are perpendicular when their direction vectors are orthogonal (dot product = 0). For lines in parametric form with y as parameter, extract direction vectors and apply the perpendicularity condition.
Step 1: Rewrite the lines in parametric form with parameter y. Line 1: x = ay + b, y = y, z = cy + d → Direction vector d_1 = (a, 1, c) Line 2: x = a'y + b', y = y, z = c'y + d' → Direction vector d_2 = (a', 1, c') Step 2: Apply perpendicularity condition: d_1 · d_2 = 0 aa' + 1(1) + cc' = 0 Step 3: Simplify to get the perpendicularity condition. aa' + cc' + 1 = 0 ∴ Answer: aa' + cc' + 1 = 0 (Option D)
Correct Answer: D