Chapter 12: Q.39 (page 332)

Vector $\overset{鈫�}{A} = 3 \hat{谋} + \hat{确}$ and vector $\overset{鈫�}{B} = 3 \hat{谋} - 2 \hat{确} + 2 \hat{k}$ . What is the cross product $\overset{鈫�}{A} 脳 \overset{鈫�}{B}$ ?

Short Answer

Expert verified

The cross product of the vectors $\overset{鈫�}{A}$ and $\overset{鈫�}{B}$ 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.

$\overset{鈫�}{A} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}$

$\overset{鈫�}{B} = b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k}$

Step :2 Explanation

The vector product of the two vectors is,

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

$A 脳 B = |\begin{array}{lll} \hat{i} 鈥呪赌呪赌呪赌� \hat{j} 鈥呪赌呪赌呪赌� \hat{k} \\ a_{1} 鈥呪赌呪赌呪赌� a_{2} 鈥呪赌呪赌呪赌� a_{3} \\ b_{1} 鈥呪赌呪赌呪赌� b_{2} 鈥呪赌呪赌呪赌� b_{3} \end{array}|$

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

Step :3 Given vectors

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

\overset{鈫�}{A} 脳 \overset{鈫�}{B} = (3 \hat{i} + \hat{j}) 脳 (3 \hat{i} - 2 \hat{j} + 2 \hat{k})

\overset{鈫�}{A} 脳 \overset{鈫�}{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 $\overset{鈫�}{A} = 3 \hat{谋} + \hat{确}$ and vector $\overset{鈫�}{B} = 3 \hat{谋} - 2 \hat{确} + 2 \hat{k}$ . What is the cross product $\overset{鈫�}{A} 脳 \overset{鈫�}{B}$ ?

Short Answer

Step by step solution

Introduction

Step :2 Explanation

Step :3 Given vectors

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Vector A鈫�=3谋^+确^ and vector B鈫�=3谋^-2确^+2k^ . What is the cross product A鈫�脳B鈫� ?

Short Answer

Step by step solution

Introduction

Step :2 Explanation

Step :3 Given vectors

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Further Mechanics and Thermal Physics

Kinematics Physics

Electrostatics

Fundamentals of Physics

Dynamics

Torque and Rotational Motion

Study anywhere. Anytime. Across all devices.

Vector $\overset{鈫�}{A} = 3 \hat{谋} + \hat{确}$ and vector $\overset{鈫�}{B} = 3 \hat{谋} - 2 \hat{确} + 2 \hat{k}$ . What is the cross product $\overset{鈫�}{A} 脳 \overset{鈫�}{B}$ ?