3D Geometry
Angle between lines
Grade None
Question:
<p>If coordinates of points <em>A</em>, <em>B</em>, <em>C</em> and <em>D</em> are (1, 2, 3); (4, 5, 7); (−4, 3, −6) and (2, 9, 2), respectively, then the angle between <em>AB</em> and <em>CD</em> is __________.</p>
Step-by-Step Solution
Key Concept: The angle between two lines in 3D space is found using the dot product of their direction vectors: cos(θ) = |d₁·d₂|/(|d₁||d₂|). We use absolute value because we measure the acute angle between lines regardless of direction.
Step 1: Find direction vectors of AB and CD. AB = B - A = (4-1, 5-2, 7-3) = (3, 3, 4) CD = D - C = (2-(-4), 9-3, 2-(-6)) = (6, 6, 8) Step 2: Calculate dot product AB·CD. AB·CD = (3)(6) + (3)(6) + (4)(8) = 18 + 18 + 32 = 68 Step 3: Calculate magnitudes. |AB| = √(3^2 + 3^2 + 4^2) = √(9 + 9 + 16) = √34 |CD| = √(6^2 + 6^2 + 8^2) = √(36 + 36 + 64) = √136 = 2√34 Step 4: Apply angle formula with absolute value. cos(θ) = |AB·CD|/(|AB||CD|) = |68|/(√34 · 2√34) = 68/(2 · 34) = 68/68 = 1 θ = cos⁻^1(1) = 0° ∴ Answer: 0° (The lines are parallel since CD = 2·AB)
Correct Answer: 0