GATE 2023 ST – Question 59

Matrix Theory · Inner Products and Quadratic Forms · 2 marks · Numerical answer

In ℝ³ with standard inner product,the orthonormal basis is v₁=(1,−1,1)ᵀ/√3,v₂=(1,2,1)ᵀ/√6,v₃=(1,0,−1)ᵀ/√2. For u=(a,b,c)ᵀ,its inner products with these vectors are 1,2,3. Find a²+b²+c² as an integer.

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Correct answer: 14 to 14

Explanation

Expand u=1v₁+2v₂+3v₃. Orthonormality eliminates cross terms,so Parseval’s identity gives ||u||²=1²+2²+3²=**14**.

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