Question

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

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

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 $(u\times v).w$.

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

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

Step 3. Finding dot product   

The required dot product is

$$\begin{gathered} (u\times v).w \\ =(-9i-7j-3k).(i+j+3k) \\ =(-9).1+(-7).1+(-3).3 \\ =-9-7-9 \\ =-25 \end{gathered}$$