Question

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

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

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 $v.(u\times w)$.

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

$$\begin{gathered} u\times w=\left|\begin{array}{ccc}i& j& k\\ 2& -3& 1\\ 1& 1& 3\end{array}\right| \\ =i\left|\begin{array}{cc}-3& 1\\ 1& 3\end{array}\right|-j\left|\begin{array}{cc}2& 1\\ 1& 3\end{array}\right|+k\left|\begin{array}{cc}2& -3\\ 1& 1\end{array}\right| \\ =i\left(\right(-3).3-1.1)-j(2.3-1.1)+k(2.1-1.(-3\left)\right) \\ =-10i-5j+5k \end{gathered}$$

Step 3. Finding dot product 

The required dot product is

$$\begin{gathered} v.(u\times w) \\ =(-3i+3j+2k).(-10i-5j+5k) \\ =(-3).(-10)+3.(-5)+2.5 \\ =30-15+10 \\ =25 \end{gathered}$$