Question

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

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

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 $u\times (v\times v)$.

Step 2. Finding first cross product  

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

The required cross product is

$$\begin{gathered} v\times v=\left|\begin{array}{ccc}i& j& k\\ -3& 3& 2\\ -3& 3& 2\end{array}\right| \\ =i\left|\begin{array}{cc}3& 2\\ 3& 2\end{array}\right|-j\left|\begin{array}{cc}-3& 2\\ -3& 2\end{array}\right|+k\left|\begin{array}{cc}-3& 3\\ -3& 3\end{array}\right| \\ =i(3.2-3.2)-j\left(\right(-3).2-(-3).2)+k\left(\right(-3).3-(-3).3) \\ =0 \end{gathered}$$

$$\begin{gathered} \therefore u\times (v\times v) \\ =u\times 0 \\ =0 \end{gathered}$$