Question

The direction ratios of a normal to the plane $x + y = 3$ can be -

• A. 0,1,0
• B. 1,1,$\sqrt{2}$
• C. 1,0,0
• D. $\sqrt{2}$,1,1

Answer 1

None of the provided options are correct. The direction ratios of a normal to the plane x + y = 3 are (1, 1, 0) or any non-zero multiple of it. None of the options satisfy this condition.

Answer 2

The equation of the plane is given as $x + y = 3$. This can be written in the standard form $Ax + By + Cz + D = 0$ as $1x + 1y + 0z - 3 = 0$.

The direction ratios of a normal vector to the plane $Ax + By + Cz + D = 0$ are $(A, B, C)$.

For the plane $x + y - 3 = 0$, the coefficients are $A=1$, $B=1$, and $C=0$.

So, the direction ratios of a normal to the plane are $(1, 1, 0)$.

Any non-zero multiple of $(1, 1, 0)$, i.e., $(k, k, 0)$ where $k \neq 0$, also represents the direction ratios of a normal to the plane.

We need to check which of the given options is proportional to $(1, 1, 0)$.

(A) (0, 1, 0) - Not proportional to (1, 1, 0). (B) (1, 1, $\sqrt{2}$) - Not proportional to (1, 1, 0). (C) (1, 0, 0) - Not proportional to (1, 1, 0). (D) ($\sqrt{2}$, 1, 1) - Not proportional to (1, 1, 0).

Therefore, none of the options are correct. There might be an error in the question or the options provided.