Question

Use the given vectors u, v, and w to find each expression.

$$\begin{gathered} u=2i-3j+k,v=-3i+3j+2k,w=i+j+3k \\ (v\times u).w \end{gathered}$$

Short answer

The required value of the expression is$25$.

Step-by-step solution

Step 1. Given Information  

We are given three vectors $u,v,w$ and we need to find the cross product of $(v\times u).w$.

Step 2. Finding cross product  

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

The required cross product is

$$\begin{gathered} v\times u=\left|\begin{array}{ccc}i& j& k\\ -3& 3& 2\\ 2& -3& 1\end{array}\right| \\ =i\left|\begin{array}{cc}3& 2\\ -3& 1\end{array}\right|-j\left|\begin{array}{cc}-3& 2\\ 2& 1\end{array}\right|+k\left|\begin{array}{cc}-3& 3\\ 2& -3\end{array}\right| \\ =i(3.1-(-3).2)-j\left(\right(-3).1-2.2)+k\left(\right(-3).(-3)-2.3) \\ =9i+7j+3k \end{gathered}$$

Step 3. Finding dot product    

The required dot product is

$$\begin{gathered} (v\times u).w \\ =(9i+7j+3k).(i+j+3k) \\ =9.1+7.1+3.3 \\ =25 \end{gathered}$$