Question

Vector $\overrightarrow{A}=3\hat{ı}+\hat{ȷ}$ and vector $\overrightarrow{B}=3\hat{ı}-2\hat{ȷ}+2\hat{k}$ . What is the cross product $\overrightarrow{A}\times \overrightarrow{B}$ ?

Short answer

The cross product of the vectors $\overrightarrow{A}$and $\overrightarrow{B}$is $2\hat{i}-6\hat{j}-9\hat{k}$.

Step-by-step solution

Introduction

Use the expression for cross product of two vectors to solve the given problem. Consider two vectors which are in terms of $\hat{i},\hat{j}$and $\hat{k}$as follows.

$\overrightarrow{A}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$

$\overrightarrow{B}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$

Step :2 Explanation

The vector product of the two vectors is,

$A\times B=\left|\begin{array}{lll}\hat{i}& \hat{j}& \hat{k}\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|$

$A\times B=\left|\begin{array}{l}\widehat{i}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\widehat{j}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\widehat{k}\\ a_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{a}_2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{a}_3\\ b_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{b}_2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{b}_3\end{array}\right|$

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

Step :3 Given vectors 

The cross product of the given vectors$\overrightarrow{A}=3\hat{i}+\hat{j}$and $\overrightarrow{B}=3\hat{i}-2\hat{j}+2\hat{k}$is expressed as follows.

$\overrightarrow{A}\times \overrightarrow{B}=(3\hat{i}+\hat{j})\times (3\hat{i}-2\hat{j}+2\hat{k})$$\overrightarrow{A}\times \overrightarrow{B}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 3& 1& 0\\ 3& -2& 2\end{array}\right|$$=\left[\right(1\left)\right(2)-(0\left)\right(-2\left)\right]\hat{i}+\left[\right(0\left)\right(3)-(3\left)\right(2\left)\right]\hat{j}+\left[\right(3\left)\right(-2)-(1\left)\right(3\left)\right]\hat{k}$$=2\hat{i}-6\hat{j}-9\hat{k}$