Chapter 10.3, Problem 31E

To determine

To evaluate: The expression $u\cdot \left(u\times v\right)$ if $u=3i-j+4k$ and $v=2i+2j-k$ .

Answer to Problem 31E

The solution of the expression $u\cdot \left(u\times v\right)$ is 0.

Explanation of Solution

Given information:

The given vectors are $u=3i-j+4k$ and $v=2i+2j-k$ .

Calculation:

Calculate the cross product of $u$ and $v$ .

  $\begin{array}{c}u\times v=\left|\begin{array}{ccc}i& j& k\\ 3& -1& 4\\ 2& 2& -1\end{array}\right|\\ =\left|\begin{array}{cc}-1& 4\\ 2& -1\end{array}\right|i-\left|\begin{array}{cc}3& 4\\ 2& -1\end{array}\right|j+\left|\begin{array}{cc}3& -1\\ 2& 2\end{array}\right|k\\ =\left(1-8\right)i-\left(-3-8\right)j+\left(6+2\right)k\\ =-7i+11j+8k\text{or}\langle -7,11,8\rangle \end{array}$

So, the cross product $u\times v$ is $-13i+60j-51k$ .

Find the dot product of vector $u$ with the cross product.

  $\begin{array}{c}u\cdot \left(u\times v\right)=\left(3i-j+4k\right)\cdot \left(-7i+11j+8k\right)\\ =-21-11+32\\ =-32+32\\ =0\end{array}$

Therefore, the solution of the expression $u\cdot \left(u\times v\right)$ is 0.