Question

Find a vector orthogonal to both u and w.

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

Short answer

The vector $10i-5j+5k$is orthogonal to both u and w

Step-by-step solution

Step1. Given information 

vectors

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

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

so orthogonal vector =$u\times w$

Step2. Calculating the orthogonal vector 

$$\begin{gathered} u=2i-3j+k \\ w=i+j+3k \\ u\times w=\left|\begin{array}{ccc}i& j& k\\ 2& -3& 1\\ 1& 1& 3\end{array}\right| \\ u\times w=[i\left|\begin{array}{cc}-3& 1\\ 1& 3\end{array}\right|-j\left|\begin{array}{cc}2& 1\\ 1& 3\end{array}\right|+k\left|\begin{array}{cc}2& -3\\ 1& 1\end{array}\right|] \\ u\times w=-10i-5j+5k \end{gathered}$$