Question

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

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

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 $u.(v\times w)$.

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

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

Step 3. Finding dot product  

The required dot product is

$$\begin{gathered} u.(v\times w) \\ =(2i-3j+k).(7i+11j-6k) \\ =2.7+(-3).11+1.(-6) \\ =14-33-6 \\ =-25 \end{gathered}$$