Chapter 1.2, Problem 36E

The dot product of two vectors $\stackrel{\to }{x}=\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]$ and $\stackrel{\to }{y}=\left[\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_n\end{array}\right]$ in ${ℝ}^{″}$ is defined by $\stackrel{\to }{x}\cdot \stackrel{\to }{y}=x_1{y}_1+x_2{y}_2+\cdots +x_n{y}_n$ .Note that the dot product of two vectors is a scalar.We say that the vectors $\stackrel{\to }{x}$ and $\stackrel{\to }{y}$ are perpendicular if $\stackrel{\to }{x}\cdot \stackrel{\to }{y}=0$ .

Find all vectors in ${ℝ}^3$ perpendicular to $\left[\begin{array}{c}1\\ 3\\ -1\end{array}\right]$ .

Draw a sketch.