Chapter 10, Problem 29P

(a)

To determine

To find: $\overrightarrow{A}\times \overrightarrow{B}$

Answer to Problem 29P

  $\overrightarrow{A}\times \overrightarrow{B}=24\widehat{k}$

Explanation of Solution

Given:

  $\begin{array}{l}\overrightarrow{A}=4\widehat{i}\\ \overrightarrow{B}=6\widehat{i}+6\widehat{j}\end{array}$

Formula used:

The property of vector cross product:

  $\begin{array}{l}\widehat{i}\times \widehat{j}=\widehat{k},\widehat{j}\times \widehat{k}=\widehat{i},\widehat{k}\times \widehat{i}=\widehat{j}\\ \widehat{i}\times \widehat{i}=0,\widehat{j}\times \widehat{j}=0,\widehat{k}\times \widehat{k}=0\end{array}$

Calculation:

  $\begin{array}{c}\overrightarrow{A}\times \overrightarrow{B}=4\widehat{i}\times \left(6\widehat{i}+6\widehat{j}\right)\\ =4\times 6\left(\widehat{i}\times \widehat{i}\right)+4\times 6\left(\widehat{i}\times \widehat{j}\right)\\ =24\times 0+24\widehat{k}\\ =24\widehat{k}\end{array}$

Conclusion:

  $\overrightarrow{A}\times \overrightarrow{B}=24\widehat{k}$

(b)

To determine

To Find: $\overrightarrow{A}\times \overrightarrow{B}$

Answer to Problem 29P

  $\overrightarrow{A}\times \overrightarrow{B}=-24\widehat{j}$

Explanation of Solution

Given:

  $\begin{array}{l}\overrightarrow{A}=4\widehat{i}\\ \overrightarrow{B}=6\widehat{i}+6\widehat{k}\end{array}$

Formula used:

The property of vector cross product:

  $\begin{array}{l}·\widehat{i}\times \widehat{j}=\widehat{k},\widehat{j}\times \widehat{k}=\widehat{i},\widehat{k}\times \widehat{i}=\widehat{j}\\ ·\widehat{j}\times \widehat{i}=-\widehat{k},\widehat{k}\times \widehat{j}=-\widehat{i},\widehat{i}\times \widehat{k}=-\widehat{j}\\ ·\widehat{i}\times \widehat{i}=0,\widehat{j}\times \widehat{j}=0,\widehat{k}\times \widehat{k}=0\end{array}$

Calculation:

  $\begin{array}{c}\overrightarrow{A}\times \overrightarrow{B}=4\widehat{i}\times \left(6\widehat{i}+6\widehat{k}\right)\\ =4\times 6\left(\widehat{i}\times \widehat{i}\right)+4\times 6\left(\widehat{i}\times \widehat{k}\right)\\ =24\times 0+24\left(\text{- }\widehat{j}\right)\\ =-24\widehat{j}\end{array}$

Conclusion:

  $\overrightarrow{A}\times \overrightarrow{B}=-24\widehat{j}$

(c)

To determine

To Find: $\overrightarrow{A}\times \overrightarrow{B}$

Answer to Problem 29P

  $\overrightarrow{A}\times \overrightarrow{B}=-5\widehat{k}$

Explanation of Solution

Given:

  $\begin{array}{l}\overrightarrow{A}=2\widehat{i}+3\widehat{j}\\ \overrightarrow{B}=3\widehat{i}+2\widehat{j}\end{array}$

Formula used:

The property of vector cross product:

  $\begin{array}{l}·\widehat{i}\times \widehat{j}=\widehat{k},\widehat{j}\times \widehat{k}=\widehat{i},\widehat{k}\times \widehat{i}=\widehat{j}\\ ·\widehat{j}\times \widehat{i}=-\widehat{k},\widehat{k}\times \widehat{j}=-\widehat{i},\widehat{i}\times \widehat{k}=-\widehat{j}\\ ·\widehat{i}\times \widehat{i}=0,\widehat{j}\times \widehat{j}=0,\widehat{k}\times \widehat{k}=0\end{array}$

Calculation:

  $\begin{array}{c}\overrightarrow{A}\times \overrightarrow{B}=\left(2\widehat{i}+3\widehat{j}\right)\times \left(3\widehat{i}+2\widehat{j}\right)\\ =2\times 3\left(\widehat{i}\times \widehat{i}\right)+2\times 2\left(\widehat{i}\times \widehat{j}\right)+3\times 3\left(\widehat{j}\times \widehat{i}\right)+3\times 2\left(\widehat{j}\times \widehat{j}\right)\\ =6\times 0+4\widehat{k}+9\left(-\widehat{k}\right)+6\times 0\\ =4\widehat{k}-9\widehat{k}\\ \overrightarrow{A}\times \overrightarrow{B}=-5\widehat{k}\end{array}$

Conclusion:

  $\overrightarrow{A}\times \overrightarrow{B}=-5\widehat{k}$