Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

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Short Answer

(a) Find the magnitude of cross product\(|a \times b|\).
(b) Check whether the components of \(a \times b\) are positive, negative or 0.

(a) The value of the cross product \(|a \times b|\) is 6.

(b) The \(x\) component of \({\rm{a}} \times {\rm{b}}\) is a positive, \(y\) component of \({\rm{a}} \times {\rm{b}}\) is a negative, and \(z\) component of \({\rm{a}} \times {\rm{b}}\) is zero.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Formula used

If \(\theta \) is the angle between vectors \({\bf{u}}\) and \({\bf{v}}\), then the cross product is,

\(|u \times v| = |u||v|\sin \theta \ldots \ldots \ldots (1)\)

Here,

|u| is the magnitude of vector\({\bf{u}}\) and

|v| is the magnitude of vector\({\bf{v}}\).

Right-hand rule:

The fingers of the right hand curl in the direction of rotation from \(a\) to \(b\), then the thumb points are in the direction of \(a \times b\). The rotation angle should be less than\({180^^\circ }\).

Step 2: Find the \(|a \times b|\)

(a)

As given, \(|{\rm{a}}| = 3,|\;{\rm{b}}| = 2\).

So, the value of angle \((\theta )\) is \({90^^\circ }\)by applying the right hand rule

Substitute 3 for \(|{\rm{a}}|,2\) for \(|{\rm{b}}|\), and \({90^^\circ }\) for \(\theta \) in \(|{\rm{a}} \times {\rm{b}}| = |{\rm{a}}||{\rm{b}}|\sin \theta \)

\(\begin{array}{l}|a \times b| = (3)(2)\sin \left( {{{90}^^\circ }} \right)\\|a \times b| = 6(1)\\|a \times b| = 6\end{array}\)

Thus, the magnitude of cross product \(|a \times b|\) is 6.

Step 3: Check whether the components of \(a \times b\) are positive, negative or 0

(b)

From part (a), it is observed that, as vector a lies on the x y- plane, the \(z\)-component of \(a \times b\)is zero.

When the direction of vector \({\rm{a}} \times {\rm{b}}\) is opposite to the direction of \(y\) axis, the \(y\)-component of \({\rm{a}} \times {\rm{b}}\) is negative.

As the direction of vector \({\rm{a}} \times {\rm{b}}\) is in the direction of \(x\) axis, the \(x\)-component of \({\rm{a}} \times {\rm{b}}\) is positive.

Therefore, the \(x\) component of \({\rm{a}} \times {\rm{b}}\) is a positive, \(y\) component of \({\rm{a}} \times {\rm{b}}\) is a negative, and \(z\) component of \(a \times b\) is zero.

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Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

(a) Find the magnitude of cross product\(|a \times b|\).
(b) Check whether the components of \(a \times b\) are positive, negative or 0.

Step by Step Solution

Step 1: Formula used

Step 2: Find the \(|a \times b|\)

Step 3: Check whether the components of \(a \times b\) are positive, negative or 0

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Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

(a) Find the magnitude of cross product\(|a \times b|\).(b) Check whether the components of \(a \times b\) are positive, negative or 0.

Step by Step Solution

Step 1: Formula used

Step 2: Find the \(|a \times b|\)

Step 3: Check whether the components of \(a \times b\) are positive, negative or 0

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(a) Find the magnitude of cross product\(|a \times b|\).
(b) Check whether the components of \(a \times b\) are positive, negative or 0.