Chapter 11, Problem 1P

To determine

The vector product $\overrightarrow{M}\times \overrightarrow{N}$.

Answer to Problem 1P

The vector product $\overrightarrow{M}\times \overrightarrow{N}$ is $1\widehat{i}+8\widehat{j}+22\widehat{k}$.

Explanation of Solution

The vector $\overrightarrow{M}$ is $2\widehat{i}-3\widehat{j}+\widehat{k}$ and the vector $\overrightarrow{N}$ is $4\widehat{i}+5\widehat{j}-2\widehat{k}$.

The cross product of the two vectors is written as,

    $\begin{array}{c}\overrightarrow{M}\times \overrightarrow{N}=\left(2\widehat{i}-3\widehat{j}+\widehat{k}\right)\times \left(4\widehat{i}+5\widehat{j}-2\widehat{k}\right)\\ =\left[\begin{array}{l}\left(2\widehat{i}\times 4\widehat{i}\right)+\left(2\widehat{i}\times 5\widehat{j}\right)+\left(2\widehat{i}\times \left(-2\widehat{k}\right)\right)+\left(-3\widehat{j}\times 4\widehat{i}\right)+\left(-3\widehat{j}\times 5\widehat{j}\right)\\ +\left(-3\widehat{j}\times \left(-2\widehat{k}\right)\right)+\left(\widehat{k}\times 4\widehat{i}\right)+\left(\widehat{k}\times 5\widehat{j}\right)+\left(\widehat{k}\times \left(-2\widehat{k}\right)\right)\end{array}\right]\\ =\left[\begin{array}{l}\left(0\right)+\left(10\widehat{k}\right)+\left(4\widehat{j}\right)+\left(12\widehat{k}\right)+\left(0\right)\\ +\left(6\widehat{i}\right)+\left(4\widehat{j}\right)+\left(-5\widehat{i}\right)+\left(0\right)\end{array}\right]\\ =1\widehat{i}+8\widehat{j}+22\widehat{k}\end{array}$

Conclusion:

Therefore, the vector product $\overrightarrow{M}\times \overrightarrow{N}$ is $1\widehat{i}+8\widehat{j}+22\widehat{k}$.