Chapter 10.3, Problem 27E

To determine

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

Answer to Problem 27E

The value of expression is $u\times \left(v\times v\right)=0$ .

Explanation of Solution

Given information:

The given expression is $u\times \left(v\times v\right)$ . The given vectors are $u=3i-j+4k$ and $v=2i+2j-k$ .

Calculation:

Write the formula of cross product of two vectors $a=a_{1}i+a_{2}j+a_{3}k$ and $b=b_{1}i+b_{2}j+b_{3}k$ .

  $a\times b=\left|\begin{array}{ccc}i& j& k\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|$ .

Or,

  $a\times b=\left(a_2{b}_3-a_3{b}_2\right)i-\left(a_1{b}_3-a_3{b}_1\right)j+\left(a_1{b}_2-a_2{b}_1\right)k$ .

Substitute the values and calculate $v\times v$ .

  $\begin{array}{c}v\times v=\left(2\times -1-\left(-1\right)\times 2\right)i-\left(2\times -1-\left(-1\right)\times 2\right)j+\left\{2\times 2-2\times 2\right\}k\\ =0i+0j+0k\end{array}$ .

Further use the above formula and solve to find $u\times \left(v\times v\right)$ .

  $\begin{array}{c}u\times \left(v\times v\right)=\left|\begin{array}{ccc}i& j& k\\ 3& -1& 4\\ 0& 0& 0\end{array}\right|\\ =0\end{array}$

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