Question

Find a vector orthogonal to both u and j + k.

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

Short answer

The orthogonal vector is : $-4i-2j+2k$

Step-by-step solution

Step1. Given information 

vectors

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

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

$$\begin{gathered} u=2i-3j+k \\ p=j+k \\ u\times p=\left|\begin{array}{ccc}i& j& k\\ 2& -3& 1\\ 0& 1& 1\end{array}\right| \\ u\times p=\left[i+\left|\begin{array}{cc}-3& 1\\ 1& 1\end{array}\right|-j\left|\begin{array}{cc}2& 1\\ 0& 1\end{array}\right|+k\left|\begin{array}{cc}2& -3\\ 0& 1\end{array}\right|\right] \\ u\times p=-4i-2j+2k \end{gathered}$$