Question

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

$u=2i-3j+k,v=-3i+3j+2k,w=i+j+3k$

$\left(3u\right)\times v$

Short answer

The required cross product is$-27i-21j-9k$.

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 $\left(3u\right)\times v$.

Step 2. Finding cross product 

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

$$\begin{gathered} \therefore 3u=3\times (2i-3j+k) \\ \Rightarrow 3u=6i-9j+3k \end{gathered}$$

The required cross product is

$$\begin{gathered} \left(3u\right)\times v=\left|\begin{array}{ccc}i& j& k\\ 6& -9& 3\\ -3& 3& 2\end{array}\right| \\ =i\left|\begin{array}{cc}-9& 3\\ 3& 2\end{array}\right|-j\left|\begin{array}{cc}6& 3\\ -3& 2\end{array}\right|+k\left|\begin{array}{cc}6& -9\\ -3& 3\end{array}\right| \\ =i\left(\right(-9).2-3.3)-j(6.2-(-3).3)+k(6.3-(-3).(-9\left)\right) \\ =-27i-21j-9k \end{gathered}$$