GATE 2016 EC (EC3) – Question 37
An orthogonal basis of $\mathbb R^3$ is $e_1=(1,0,2)$, $e_2=(0,1,0)$, $e_3=(-2,0,1)$. Express $u=(4,3,-3)$ in this basis.
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Correct answer: (D) $-\tfrac25e_1+3e_2-\tfrac{11}5e_3$
Explanation
In an orthogonal basis, the coefficient of ei is $(u\cdot e_i)/(e_i\cdot e_i)$.
The dot products are −2,3,−11 and the squared basis lengths are 5,1,5. The coefficients are therefore −2/5,3,−11/5.
Thus **$u=-\tfrac25e_1+3e_2-\tfrac{11}5e_3$, D**. Substitution recovers (4,3,−3).