Question

Find a vector orthogonal to both u and v.

$$\begin{gathered} u=2i-3j+k \\ v=-3i+3j+2k \end{gathered}$$

Short answer

The vector $-9i-7j-3k$is orthogonal to both u and v

Step-by-step solution

Step1. Given information 

vectors $$\begin{gathered} u=2i-3j+k \\ v=-3i+3j+2k \end{gathered}$$

According to geometrical properties of the cross, the product is the resultant of the cross product is orthogonal to both vectors

so orthogonal vector = $u\times v$

Step2.Calculating the cross product 

$$\begin{gathered} u\times v=\left|\begin{array}{ccc}i& j& k\\ 2& -3& 1\\ -3& 3& 2\end{array}\right| \\ u\times v=i\left|\begin{array}{cc}-3& \\ 3& 2\end{array}\right|-j\left|\begin{array}{cc}2& 1\\ -3& 2\end{array}\right|+k\left|\begin{array}{cc}2& -3\\ -3& 3\end{array}\right| \\ u\times v=-9i-7j-3k \end{gathered}$$