Question

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

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

Short answer

The required value of the expression is $0$.

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 $v.(v\times w)$.

Step 2. Finding cross product  

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

The required cross product is

$$\begin{gathered} v\times w=\left|\begin{array}{ccc}i& j& k\\ -3& 3& 2\\ 1& 1& 3\end{array}\right| \\ =i\left|\begin{array}{cc}3& 2\\ 1& 3\end{array}\right|-j\left|\begin{array}{cc}-3& 2\\ 1& 3\end{array}\right|+k\left|\begin{array}{cc}-3& 3\\ 1& 1\end{array}\right| \\ =i(3.3-1.2)-j\left(\right(-3).3-1.2)+k\left(\right(-3).1-1.3) \\ =7i+11j-6k \end{gathered}$$

Step 3. Finding dot product 

The required dot product is

$$\begin{gathered} v.(v\times w) \\ =(-3i+3j+2k).(7i+11j-6k) \\ =(-3).7+3.11+2.(-6) \\ =-21+33-12 \\ =0 \end{gathered}$$