GATE 2025 EC – Question 59
Consider the vectors
$$\boldsymbol a=\begin{bmatrix}1\\1\end{bmatrix},\qquad\boldsymbol b=\begin{bmatrix}0\\3\sqrt2\end{bmatrix}.$$
For real-valued scalar variable $x$, the value of
$$\min_x\|\boldsymbol ax-\boldsymbol b\|_2$$
is ____________ (rounded off to two decimal places).
$\|\cdot\|_2$ denotes the Euclidean norm, i.e., for $\boldsymbol y=\begin{bmatrix}y_1\\y_2\end{bmatrix}$, $\|\boldsymbol y\|_2=\sqrt{y_1^2+y_2^2}$.
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Show answer and explanation
Correct answer: 2.98 to 3.02
Explanation
The minimum is the distance from $\boldsymbol b$ to the line through $\boldsymbol a$. The best $x=\dfrac{\boldsymbol a^T\boldsymbol b}{\boldsymbol a^T\boldsymbol a}=\dfrac{3\sqrt2}{2}$. The residual is $\boldsymbol b-x\boldsymbol a=\left(-\tfrac{3\sqrt2}{2},\ \tfrac{3\sqrt2}{2}\right)$, whose norm is $\sqrt{2\cdot\tfrac{18}{4}}=3$.