Question

Find a vector orthogonal to both u and $i+j$

$u=2i-3j+kv=-3i+3j+2kw=i+j+3k$

Short answer

The orthogonal vector is :$-i+j+5k$

Step-by-step solution

Step1. Given information 

vectors

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

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

so orthogonal vector =$u\times p$

Step2. Calculating the cross product of u and p 

$$\begin{gathered} u=2i-3j+k \\ p=i+j \\ u\times p=\left|\begin{array}{ccc}i& j& k\\ 2& -3& 1\\ 1& 1& 0\end{array}\right| \\ u\times p=\left[i+\left|\begin{array}{cc}-3& 1\\ 1& 0\end{array}\right|-j\left|\begin{array}{cc}2& 1\\ 1& 0\end{array}\right|+k\left|\begin{array}{cc}2& -3\\ 1& 1\end{array}\right|\right] \\ u\times p=-i+j+5k \end{gathered}$$