Chapter 9.5, Problem 17E

To determine

To find: The perpendicular vector to the plane passing through the given points.

Answer to Problem 17E

The perpendicular vector to the plane passing through the given pointsis $2j+2k$ .

Explanation of Solution

Given information:

The points are $P\left(0,1,0\right),Q\left(1,2,-1\right),R\left(-2,1,0\right)$ .

Calculation:

The length of the vector $\overrightarrow{PQ}$ is,

  $\begin{array}{c}\overrightarrow{PQ}=\left(1-0\right)i+\left(2-1\right)j+\left(-1-0\right)k\\ =i+j-k\end{array}$

The length of the vector $\overrightarrow{PR}$ is,

  $\begin{array}{c}\overrightarrow{PR}=\left(-2-0\right)i+\left(1-0\right)j+\left(0-0\right)k\\ =-2i\end{array}$

The cross product is given as,

  $a\times b=\left|\begin{array}{ccc}i& j& k\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|$

Substitute the values in above equation.

  $\begin{array}{c}\overrightarrow{PQ}\times \overrightarrow{PR}=\left|\begin{array}{ccc}i& j& k\\ 1& 1& -1\\ -2& 0& 0\end{array}\right|\\ =2j+2k\end{array}$