Chapter 11.3, Problem 32E

To determine

ToFind:The unit vector that is orthogonal tovectors $u=9i+3j-12k$ and $v=-\frac{2}{3}i-\frac{1}{3}j+\frac{1}{3}k$

Answer to Problem 32E

The unit vector which is orthogonal to both given vectors $u$ and $v$ is $\frac{u\times v}{\Vert u\times v\Vert }=\frac{5\sqrt{51}i}{51}+\frac{5\sqrt{51}j}{51}-\frac{\sqrt{51}k}{51}$

Explanation of Solution

Given:

The vectors $u=9i+3j-12k$ and $v=-\frac{2}{3}i-\frac{1}{3}j+\frac{1}{3}k$

Concept Used:

For the two vectors $u=\langle u_1,u_2,u_3\rangle$ and $v=\langle v_1,v_2,v_3\rangle$ , the vector which is orthogonal to both vectors $u$ and $v$ can be calculated as

  $u\times v=\left|\begin{array}{ccc}i& j& k\\ u_1& u_2& u_3\\ v_1& v_2& v_3\end{array}\right|$

The unit vector which is orthogonal to both vectors $u$ and $v$ is given by $\frac{u\times v}{\Vert u\times v\Vert }$

Calculation:

Forthe vectors $u=9i+3j-12k$ and $v=-\frac{2}{3}i-\frac{1}{3}j+\frac{1}{3}k$

  $\begin{array}{c}u\times v=\left|\begin{array}{ccc}i& j& k\\ 9& 3& -12\\ -\frac{2}{3}& -\frac{1}{3}& \frac{1}{3}\end{array}\right|\\ =i\left|\begin{array}{cc}3& -12\\ -\frac{1}{3}& \frac{1}{3}\end{array}\right|-j\left|\begin{array}{cc}9& -12\\ -\frac{2}{3}& \frac{1}{3}\end{array}\right|+k\left|\begin{array}{cc}9& 3\\ -\frac{2}{3}& -\frac{1}{3}\end{array}\right|\\ =i\left(\left(3\right)\left(\frac{1}{3}\right)-\left(-\frac{1}{3}\right)\left(-12\right)\right)-j\left(\left(9\right)\left(\frac{1}{3}\right)-\left(-\frac{2}{3}\right)\left(-12\right)\right)+k\left(\left(9\right)\left(-\frac{1}{3}\right)-\left(-\frac{2}{3}\right)\left(3\right)\right)\\ =i\left(1+4\right)-j\left(3-8\right)+k\left(-3+2\right)\end{array}$

  $=5i+5j-1k$

Therefore, $u\times v=5i+5j-1k$

The length of $u\times v$ is

  $\begin{array}{c}\Vert u\times v\Vert =\sqrt{{\left(5\right)}^2+{\left(5\right)}^2+{\left(-1\right)}^2}\\ =\sqrt{25+25+1}\\ =\sqrt{51}\end{array}$

The unit vector is

  $\begin{array}{c}\frac{u\times v}{\Vert u\times v\Vert }=\frac{5i+5j-1k}{\sqrt{51}}\\ =\frac{5i}{\sqrt{51}}+\frac{5j}{\sqrt{51}}-\frac{1k}{\sqrt{51}}\\ =\frac{5\sqrt{51}i}{51}+\frac{5\sqrt{51}j}{51}-\frac{\sqrt{51}k}{51}\end{array}$

Therefore, unit vector which is orthogonal to both given vectors $u$ and $v$ is $\frac{u\times v}{\Vert u\times v\Vert }=\frac{5\sqrt{51}i}{51}+\frac{5\sqrt{51}j}{51}-\frac{\sqrt{51}k}{51}$

Conclusion:

The unit vector which is orthogonal to both given vectors $u$ and $v$ is $\frac{u\times v}{\Vert u\times v\Vert }=\frac{5\sqrt{51}i}{51}+\frac{5\sqrt{51}j}{51}-\frac{\sqrt{51}k}{51}$