GATE 2023 EE Q.11: solution and answer

Official answer: DSource: Official answer key, GATE 2023 EE (IIT Kanpur)

1. Write the plane out. With $\mathbf{x}=[x_1{x}_2{x}_3{]}^T$, the equation ${\mathbf{w}}^T\mathbf{x}=1$ is $x_1+2x_2+3x_3=1$.

2. Why $\mathbf{w}$ is normal to this plane: take any two points $\mathbf{p}$ and $\mathbf{q}$ on the plane. Then ${\mathbf{w}}^T\mathbf{p}=1$ and ${\mathbf{w}}^T\mathbf{q}=1$, so $${\mathbf{w}}^T(\mathbf{p}-\mathbf{q})=1-1=0.$$ Every direction lying in the plane has the form $\mathbf{p}-\mathbf{q}$, so $\mathbf{w}$ is orthogonal to all of them. That makes $\mathbf{w}$ normal to the plane.

3. So the normal is $\mathbf{w}=[123{]}^T$ or any nonzero multiple of it. The constant on the right-hand side (here $1$) only moves the plane parallel to itself. It does not change the normal direction.

4. Test the other options with the dot product. For (A), $[-2-22]\mathbf{w}=-2-4+6=0$. For (B), $[30-1]\mathbf{w}=3+0-3=0$. Both are orthogonal to $\mathbf{w}$, so they are directions parallel to the plane, not normal to it.

5. Option (C), $[321{]}^T$, is not a scalar multiple of $[123{]}^T$ because the component ratios $3/1$, $2/2$ and $1/3$ are not equal. Its dot product with $\mathbf{w}$ is $3+4+3=10\ne 0$, so it is not parallel to the plane either. It is neither normal nor parallel to the plane.

6. Only option (D), $[123{]}^T=\mathbf{w}$, is normal to the plane. Answer: (D).