Question

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

$$\begin{gathered} u=2i-3j+k,v=-3i+3j+2k,w=i+j+3k \\ v\times \left(4w\right) \end{gathered}$$

Short answer

The required cross product is$28i+44j-24k$.

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$v\times \left(4w\right)$.

Step 2. Finding cross product  

$$\begin{gathered} v=-3i+3j+2k \\ w=i+j+3k \\ \therefore \left(4w\right)=4\times (i+j+3k) \\ \Rightarrow 4w=4i+4j+12k \end{gathered}$$

The required cross product is

$$\begin{gathered} v\times \left(4w\right)=\left|\begin{array}{ccc}i& j& k\\ -3& 3& 2\\ 4& 4& 12\end{array}\right| \\ =i\left|\begin{array}{cc}3& 2\\ 4& 12\end{array}\right|-j\left|\begin{array}{cc}-3& 2\\ 4& 12\end{array}\right|+k\left|\begin{array}{cc}-3& 3\\ 4& 4\end{array}\right| \\ =i(3.12-4.2)-j\left(\right(-3).12-4.2)+k\left(\right(-3).4-3.4) \\ =28i+44j-24k \end{gathered}$$