Solve the problem.
-Find a unit vector perpendicular to plane PQR determined by the points P(2, 1, 3), Q(1, 1, 2) and R(2, 2, 1).
A) $\frac { 1 } { \sqrt { 6 } } ( \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } )$
B) $\frac { 1 } { \sqrt { 6 } } ( \mathbf { i } - 2 \mathbf { j } - \mathbf { k } )$
C) $\frac { 1 } { \sqrt { 6 } } ( \mathbf { i } - 2 \mathrm { j } + \mathbf { k } )$
D) $\frac { 1 } { \sqrt { 6 } } ( - \mathbf { i } - 2 \mathbf { j } - \mathbf { k } )$

Question

Solve the problem.
-Find a unit vector perpendicular to plane PQR determined by the points P(2, 1, 3), Q(1, 1, 2) and R(2, 2, 1). 
A) $\frac { 1 } { \sqrt { 6 } } ( \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } )$
B) $\frac { 1 } { \sqrt { 6 } } ( \mathbf { i } - 2 \mathbf { j } - \mathbf { k } )$
C) $\frac { 1 } { \sqrt { 6 } } ( \mathbf { i } - 2 \mathrm { j } + \mathbf { k } )$
D) $\frac { 1 } { \sqrt { 6 } } ( - \mathbf { i } - 2 \mathbf { j } - \mathbf { k } )$

Quizplus · Accepted Answer

To find a unit vector perpendicular to the plane PQR, we first need to find two vectors in the plane. We can choose the vectors PQ and PR, which are (1, 0, -1) and (0, 1, -2), respectively. Then, we can find the cross product of these two vectors to get a vector that is perpendicular to both of them, which will also be perpendicular to the plane.

\begin{align*} PQ 	imes PR &= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 0 & -1 \ 0 & 1 & -2 \end{vmatrix} \ &= (-1\mathbf{i} + 2\mathbf{j} + 1\mathbf{k}) \end{align*}

Next, we need to find a unit vector in the same direction as this vector. To do this, we divide the vector by its magnitude:

\begin{align*} \| (-1\mathbf{i} + 2\mathbf{j} + 1\mathbf{k}) \| &= \sqrt{(-1)^2 + 2^2 + 1^2} \ &= \sqrt{6} \end{align*}

Therefore, a unit vector in the direction of PQ × PR is:

\begin{align*} \frac{1}{\sqrt{6}}(-1\mathbf{i} + 2\mathbf{j} + 1\mathbf{k}) &= \frac{1}{\sqrt{6}}(\mathbf{i} - 2\mathbf{j} - \mathbf{k}) \end{align*}

The closest answer choice to this vector is B, so the answer is B.

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Solve the Problem $\frac { 1 } { \sqrt { 6 } } ( \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } )$