3D Geometry
Perpendicularity of lines in 3D
Grade 12
Question:
<p>Line <em>x = ay + b, z = cy + d</em> and line <em>x = a'z + b', y = c' + d'</em> are perpendicular to each other. Then which condition holds?</p>
<p><em>aa' + cc' = 0</em></p>
<p><em>aa' + c + c' = 0</em></p>
<p><em>aa' + cc' + 1 = 0</em></p>
<p><em>a + a' + c + c' = 0</em></p>
Step-by-Step Solution
Key Concept: Two lines in 3D are perpendicular when the dot product of their direction vectors equals zero. Convert parametric forms to find direction vectors by expressing each line in terms of a single parameter.
Step 1: Convert Line 1 (x = ay + b, z = cy + d) to parametric form with parameter t = y:
x = at + b, y = t, z = ct + d
Direction vector: d_1 = (a, 1, c) Step 2: Convert Line 2 (x = a'z + b', y = c'z + d') to parametric form with parameter s = z:
x = a's + b', y = c's + d', z = s
Direction vector: d_2 = (a', c', 1) Step 3: For perpendicular lines, apply the condition d_1·d_2 = 0:
aa' + 1·c' + c·1 = 0
aa' + c + c' = 0 Step 4: Therefore, the perpendicularity condition is: aa' + c + c' = 0 ∴ Answer: B
Correct Answer: B