GATE 2022 ST – Question 23

Matrix Theory · Inner Products and Quadratic Forms · 1 mark · Numerical answer

A unit vector u∈ℝ⁴ is orthogonal to S={x:x₁+x₂+x₃+x₄=0}. Let v=(1,−2,1,1)ᵀ. If u and v−αu are orthogonal,find α² to two decimals.

Show answer and explanation

Correct answer: 0.25 to 0.25

Explanation

The normal unit vectors are u=±(1,1,1,1)ᵀ/2. Orthogonality gives α=u·v=±(1−2+1+1)/2=±1/2. Squaring gives **0.25**.

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