Question

What is the unit vector perpendicular to the following vectors,$2i+2j-k$ and $6i-3j+2k$.

Answer 1

Step 1:Given:

$$\begin{gathered} \stackrel{\rightharpoonup }{a}=2i+2j-k \\ \stackrel{\rightharpoonup }{b}=6i-3j+2k \end{gathered}$$

For finding unit vectors perpendicular to $\stackrel{\rightharpoonup }{a}$ and $\stackrel{\rightharpoonup }{b}$. First calculate cross product of $\stackrel{\rightharpoonup }{a}$ and $\stackrel{\rightharpoonup }{b}$ because $\stackrel{\rightharpoonup }{a}\times \stackrel{\rightharpoonup }{b}$ is perpendicular to $\stackrel{\rightharpoonup }{a}$ and $\stackrel{\rightharpoonup }{b}$.

Step 2:To Find:

Unit vector perpendicular to both the given

Step 3: Calculate the cross product of the vector

$$\begin{gathered} \stackrel{\rightharpoonup }{c}=\stackrel{\rightharpoonup }{a}\times \stackrel{\rightharpoonup }{b} \\ =\left|\begin{array}{ccc}i& j& k\\ 2& 2& -1\\ 6& -3& 2\end{array}\right| \\ =i-10j-18k \end{gathered}$$

Step 4: Unit vector

$$\begin{gathered} \hat{c}=\frac{\stackrel{\rightharpoonup }{c}}{\left|\stackrel{\rightharpoonup }{c}\right|} \\ \hat{c}=\frac{i-10j-18k}{\sqrt{1+100+324}} \\ \hat{c}=\frac{i-10j-18k}{5\sqrt{17}} \end{gathered}$$

Hence, the unit vector perpendicular to the following vectors, $2i+2j-k$ and $6i-3j+2k$is $\hat{c}=\frac{i-10j-18k}{5\sqrt{17}}$.