Chapter 10.3, Problem 34E

To determine

To find: The unit vector orthogonal to both $u$ and $v$ if $u=\langle 2,-1,3\rangle$ and $v=\langle 1,0,-2\rangle$ .

Answer to Problem 34E

The unit vector orthogonal to both $u$ and $v$ is $\frac{2}{3\sqrt{6}}\text{i+}\frac{7}{3\sqrt{6}}\text{j+}\frac{1}{3\sqrt{6}}\text{k}$ .

Explanation of Solution

Given information:

The given value are $u=\langle 2,-1,3\rangle$ and $v=\langle 1,0,-2\rangle$ .

Calculation:

Calculate the cross product $u\times v$ .

  $\begin{array}{c}u\times v=\left(\begin{array}{ccc}i& j& k\\ 2& -1& 3\\ 1& 0& -2\end{array}\right)\\ =\left(\begin{array}{cc}-1& 3\\ 0& -2\end{array}\right)i-\left(\begin{array}{cc}2& 3\\ 1& -2\end{array}\right)j+\left(\begin{array}{cc}2& -1\\ 1& 0\end{array}\right)k\\ =\left(2-0\right)i-\left(-4-3\right)j+\left(0+1\right)k\\ =2i+7j+k\end{array}$

Calculate the unit vector of $u\times v$ .

  $\begin{array}{c}\Vert u\times v\Vert =\sqrt{2^2+{\left(7\right)}^2+{\left(1\right)}^2}\\ =\sqrt{54}\\ =3\sqrt{6}\end{array}$

Calculate the unit vector orthogonal to both $u$ and $v$ .

  $\begin{array}{c}\frac{u\times v}{\Vert u\times v\Vert }=\frac{2i+7\text{j+k}}{3\sqrt{6}}\\ =\frac{2}{3\sqrt{6}}\text{i+}\frac{7}{3\sqrt{6}}\text{j+}\frac{1}{3\sqrt{6}}\text{k}\end{array}$

Therefore, the unit vector orthogonal to both $u$ and $v$ is $\frac{2}{3\sqrt{6}}\text{i+}\frac{7}{3\sqrt{6}}\text{j+}\frac{1}{3\sqrt{6}}\text{k}$ .