1)

 A unit vectorperpendicular to the plane formed by the points (1, 0, 1), (0,2,2) and (3, 3, 0) is


A) $\frac{1}{5\sqrt{3}}(5\hat{i}-\hat{j}-7\hat{k})$

B) $\frac{1}{5\sqrt{3}}(5\hat{i}-\hat{j}+7\hat{k})$

C) $\frac{1}{5\sqrt{3}}(5\hat{i}+\hat{j}+7\hat{k})$

D) None of these

Answer:

Option B

Explanation:

Let A(1, 0, 1), B(0, 2,2) andc (3, 3, 0) be the given points, then

$\vec{AB}=-\hat{i}+2\hat{j}+2\hat{k}$, $\vec{BC}=3\hat{i}+\hat{j}-2\hat{k}$

$\vec{AB}\times\vec{BC}=\begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\-1 & 2 & 1 \\ 3 & 1 & -2\end{vmatrix}= -5\hat{i}+\hat{j}-7\hat{k}$

.'. unit vector $\perp$ to the Plane: ABC

= $\pm\frac{1}{5\sqrt{3}}(-5\hat{i}+\hat{j}-7\hat{k})$