For a given vector w=[1\ 2\ 3]T, the vector normal to the plane defined by wT x=1 is

GATE 2023 EE – Question 11

Engineering Mathematics · Linear Algebra: Matrix Algebra · 1 mark · Multiple choice

For a given vector $\mathbf w=[1\ 2\ 3]^T$, the vector normal to the plane defined by $\mathbf w^T\mathbf x=1$ is

$[-2\ \ -2\ \ 2]^T$
$[3\ \ 0\ \ -1]^T$
$[3\ \ 2\ \ 1]^T$
$[1\ \ 2\ \ 3]^T$

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Correct answer: (D) $[1\ \ 2\ \ 3]^T$

Explanation

For a plane $\mathbf w^T\mathbf x=c$ the gradient of $\mathbf w^T\mathbf x$ is $\mathbf w$, which is normal to the plane. So the normal is $[1\ 2\ 3]^T$. (Options A and B are perpendicular to $\mathbf w$, i.e. lie in the plane.)