Chapter 11.3, Problem 11E

To determine

The value of the expression $u\times \left(-v\right)$ .

Answer to Problem 11E

The value of the expression $u\times \left(-v\right)$ is $7i-11j-8k$ .

Explanation of Solution

Given information:

The value of the vectors $u=3i-j+4k$ and $v=2i+2j-k$ .

Formula used:

If there are two three-dimensional vectors in a space, say, $a=a_{1}i+a_{2}j+a_{3}k$ and $b=b_{1}i+b_{2}j+b_{3}k$ , then the cross product of two vectors is defined as ,

  $\begin{array}{c}a\times b=\begin{array}{ccc}i& j& k\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\\ \text{= }\left(a_2{b}_3-a_3{b}_2\right)i-\left(a_1{b}_3-a_3{b}_1\right)j+\left(a_1{b}_2-a_2{b}_1\right)k\end{array}$

Calculation:

Consider the given value of the vectors $u=3i-j+4k$ and $v=2i+2j-k$ .

The value of term $\left(-\text{v}\right)$ will be calculated as when $\left(-1\right)$ is multiplied by the vector $v=2i+2j-k$ .

So, $\left(-\text{v}\right)$ is equal to,

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

Recall that if there are two three-dimensional vectors in a space, say, $a=a_{1}i+a_{2}j+a_{3}k$ and $b=b_{1}i+b_{2}j+b_{3}k$ , then the cross product of two vectors is defined as ,

  $\begin{array}{c}a\times b=\begin{array}{ccc}i& j& k\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\\ \text{= }\left(a_2{b}_3-a_3{b}_2\right)i-\left(a_1{b}_3-a_3{b}_1\right)j+\left(a_1{b}_2-a_2{b}_1\right)k\end{array}$

So, the cross product of $u\times \left(-v\right)$ is calculated as,

  $\begin{array}{c}u\times \left(-v\right)=\begin{array}{ccc}i& j& k\\ 3& -1& 4\\ -2& -2& 1\end{array}\\ \text{= }\left(\left[\left(-1\right)\times 1\right]-\left[4\times \left(-2\right)\right]\right)i-\left(\left[3\times 1\right]-\left[4\times \left(-2\right)\right]\right)j+\left(\left[3\times \left(-2\right)\right]-\left[\left(-2\right)\times \left(-1\right)\right]\right)k\\ \text{=}\left(-1-\left(-8\right)\right)i-\left(3-\left(-8\right)\right)j+\left(-6-2\right)k\\ \text{=}\left(-1+8\right)i-\left(3+8\right)j+\left(-8\right)k\end{array}$

Simplify it further as,

  $\begin{array}{c}u\times \left(-v\right)=\left(-1+8\right)i-\left(3+8\right)j+\left(-8\right)k\\ \text{=}7i-11j-8k\end{array}$

Hence, the value of the expression $u\times \left(-v\right)$ is $7i-11j-8k$ .