Chapter 12: Q.39 (page 332)

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

Short Answer

Expert verified

The cross product of the vectors $\vec{A}$ and $\vec{B}$ 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.

$\vec{A} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}$

$\vec{B} = b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k}$

Step :2 Explanation

The vector product of the two vectors is,

$A \times B = |\begin{array}{l} \hat{i} & \hat{j} & \hat{k} \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{array}|$

$A \times B = |\begin{array}{lll} \hat{i} \hat{j} \hat{k} \\ a_{1} a_{2} a_{3} \\ b_{1} b_{2} b_{3} \end{array}|$

$= (a_{2} b_{3} - a_{3} b_{2}) \hat{i} + (a_{3} b_{1} - a_{1} b_{3}) \hat{j} + (a_{1} b_{2} - a_{2} b_{1}) \hat{k}$

Step :3 Given vectors

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

\vec{A} \times \vec{B} = (3 \hat{i} + \hat{j}) \times (3 \hat{i} - 2 \hat{j} + 2 \hat{k})

\vec{A} \times \vec{B} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 1 & 0 \\ 3 & - 2 & 2 \end{matrix}|

= [(1) (2) - (0) (- 2)] \hat{i} + [(0) (3) - (3) (2)] \hat{j} + [(3) (- 2) - (1) (3)] \hat{k}

= 2 \hat{i} - 6 \hat{j} - 9 \hat{k}

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

Short Answer

Step by step solution

Introduction

Step :2 Explanation

Step :3 Given vectors

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Vector A→=3ı^+ȷ^ and vector B→=3ı^-2ȷ^+2k^ . What is the cross product A→×B→ ?

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Step by step solution

Introduction

Step :2 Explanation

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One App. One Place for Learning.

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