3D Geometry
Planes
Grade 12
Question:
<p>Given \(\overrightarrow{OQ} = (1-3\mu)\hat{i} + (\mu-1)\hat{j} + (5\mu+2)\hat{k}\) and \(\overrightarrow{OP} = 3\hat{i} + 2\hat{j} + 6\hat{k}\) (where O is the origin). If \(\overrightarrow{PQ}\) is parallel to the plane \(x - 4y + 3z = 1\), find the value of \(\mu\).</p>
Step-by-Step Solution
Key Concept: A vector is parallel to a plane if and only if it is perpendicular to the plane's normal vector. Use the condition that the dot product of PQ with the normal vector equals zero.
Step 1: Find vector PQ PQ = OQ - OP = [(1-3μ) - 3]î + [(μ-1) - 2]ĵ + [(5μ+2) - 6]k̂ PQ = (-2-3μ)î + (μ-3)ĵ + (5μ-4)k̂ Step 2: Identify the normal vector to the plane For plane x - 4y + 3z = 1, the normal vector is n = î - 4ĵ + 3k̂ Step 3: Apply the parallel condition For PQ to be parallel to the plane, PQ · n = 0 (-2-3μ)(1) + (μ-3)(-4) + (5μ-4)(3) = 0 Step 4: Expand and simplify -2 - 3μ - 4μ + 12 + 15μ - 12 = 0 (-3 - 4 + 15)μ + (-2 + 12 - 12) = 0 8μ - 2 = 0 8μ = 2 μ = 1/4 = 0.25 ∴ Answer: 0.25
Correct Answer: 0.25