Allied & ICAR Exam Bank • ALLIED_1049
NAT • 1 Mark
Section 1: Engineering Mathematics
Linear Algebra
Subtopic: Orthogonal vectors
Question Statement:
The value of $\alpha$ for which the vectors $\vec{u} = (2, -1, 3)$ and $\vec{v} = (1, 4, \alpha)$ are orthogonal is:
Step-by-Step Solution & Explanation
Two vectors are orthogonal if their dot product is zero: $\vec{u} \cdot \vec{v} = 2(1) + (-1)(4) + 3(\alpha) = 2 - 4 + 3\alpha = -2 + 3\alpha = 0 \implies 3\alpha = 2 \implies \alpha = 2/3 \approx 0.667$.
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