Log In Start studying!

Select your language

Suggested languages for you:

Americas

English (US)

Europe

Q13E

Expert-verified

Essential Calculus: Early Transcendentals

Found in: Page 556

Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

Answers without the blur.

Just sign up for free and you're in.

Register for Free I’ll do it later

Short Answer

To show
(a) \({\rm{i}} \cdot {\rm{j}} = 0,{\rm{j}} \cdot {\rm{k}} = 0\) and \({\rm{k}} \cdot {\rm{i}} = 0\).
(b) \({\rm{i}} \cdot {\rm{i}} = 1,{\rm{j}} \cdot {\rm{j}} = 1\) and \({\rm{k}} \cdot {\rm{k}} = 1\)

(a)\({\rm{i}} \cdot {\rm{j}} = 0,{\rm{j}} \cdot {\rm{k}} = 0\) and \({\rm{k}} \cdot {\rm{i}} = 0\)is shown.

(b) \({\rm{i}} \cdot {\rm{i}} = 1,{\rm{j}} \cdot {\rm{j}} = 1\) and \({\rm{k}} \cdot {\rm{k}} = 1\)is shown.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1: Concept of Dot Product

Formula:

Write a general expression to find the dot product between two three-dimensional vectors.

\(\begin{aligned}{l}{\rm{a}} \cdot {\rm{b}} &= \left\langle {{a_1},{a_2},{a_3}} \right\rangle \cdot \left\langle {{b_1},{b_2},{b_3}} \right\rangle \\{\rm{a}} \cdot {\rm{b}} &= {a_1}{b_1} + {a_2}{b_2} + {a_3}{b_3}\end{aligned}\)

Step 2: Calculation to represent vectors in three- dimensional form

Represent \(i,j,k\)vectors in three-dimensional form.

\(\begin{aligned}{l}{\bf{i}} &= 1{\rm{i}} + 0{\rm{j}} + 0{\rm{k}}\\{\bf{i}} &= \langle 1,0,0\rangle \\{\rm{j}} &= 0{\rm{i}} + 1{\rm{j}} + 0{\rm{k}}\\{\rm{j}} &= \langle 0,1,0\rangle \\{\rm{k}} &= 0{\rm{i}} + 0{\rm{j}} + 1{\rm{k}}\\{\rm{k}} &= \langle 0,0,1\rangle \end{aligned}\)

Step 3: Calculation for dot product of\({\rm{i}} \cdot {\rm{j}},{\rm{j}} \cdot {\rm{k}}\)and\({\rm{k}} \cdot {\rm{i}}\)

Find the dot product of \({\bf{i}}\) and \({\bf{j}}\) using formula.

\(\begin{aligned}{l}{\rm{i}} \cdot {\rm{j}} &= \langle 1,0,0\rangle \cdot \langle 0,1,0\rangle \\{\rm{i}} \cdot {\rm{j}} &= (1)(0) + (0)(1) + (0)(0)\\{\rm{i}} \cdot {\rm{j}} &= 0 + 0 + 0\\{\rm{i}} \cdot {\rm{j}} &= 0\end{aligned}\)

Find the dot product of \({\bf{j}}\) and \({\bf{k}}\) using formula.

\(\begin{aligned}{l}{\rm{j}} \cdot {\rm{k}} &= \langle 0,1,0\rangle \cdot \langle 0,0,1\rangle \\{\rm{j}} \cdot {\rm{k}} &= (0)(0) + (1)(0) + (0)(1)\\{\rm{j}} \cdot {\rm{k}} &= 0 + 0 + 0\\{\rm{j}} \cdot {\rm{k}} &= 0\end{aligned}\)

Find the dot product of \({\bf{k}}\) and \({\bf{i}}\) using formula.

\(\begin{aligned}{l}{\rm{k}} \cdot {\rm{i}} &= \langle 0,0,1\rangle \cdot \langle 1,0,0\rangle \\{\rm{k}} \cdot {\rm{i}} &= (0)(1) + (0)(0) + (1)(0)\\{\rm{k}} \cdot {\rm{i}} &= 0 + 0 + 0\\{\rm{k}} \cdot {\rm{i}} &= 0\end{aligned}\)

Thus, \({\rm{i}} \cdot {\rm{j}} = {\rm{j}} \cdot {\rm{k}} = {\rm{k}} \cdot {\rm{i}} = 0\) is shown.

Step 4: Calculation for dot product of\({\rm{i}} \cdot {\rm{i}},{\rm{j}} \cdot {\rm{j}}\)and\({\rm{k}} \cdot {\rm{k}}\)

Find the dot product of \({\bf{i}}\) and \({\bf{i}}\) using formula.

\(\begin{aligned}{l}{\rm{i}} \cdot {\rm{i}} &= \langle 1,0,0\rangle \cdot \langle 1,0,0\rangle \\{\rm{i}} \cdot {\rm{i}} &= (1)(1) + (0)(0) + (0)(0)\\{\rm{i}} \cdot {\rm{i}} &= 1 + 0 + 0\\{\rm{i}} \cdot {\rm{i}} &= 1\end{aligned}\)

Find the dot product of \({\bf{j}}\) and \({\bf{j}}\) using formula.

\(\begin{aligned}{l}{\rm{j}} \cdot {\rm{j}} &= \langle 0,1,0\rangle \cdot \langle 0,1,0\rangle \\{\rm{j}} \cdot {\rm{j}} &= (0)(0) + (1)(1) + (0)(0)\\{\rm{j}} \cdot {\rm{j}} &= 0 + 1 + 0\\{\rm{j}} \cdot {\rm{j}} &= 1\end{aligned}\)

Find the dot product of \({\bf{k}}\) and \({\bf{k}}\) using formula.

\(\begin{aligned}{l}{\rm{k}} \cdot {\rm{k}} &= \langle 0,0,1\rangle \cdot \langle 0,0,1\rangle \\{\rm{k}} \cdot {\rm{k}} &= (0)(0) + (0)(0) + (1)(1)\\{\rm{k}} \cdot {\rm{k}} &= 0 + 0 + 1\\{\rm{k}} \cdot {\rm{k}} &= 1\end{aligned}\)

Thus, \({\rm{i}} \cdot {\rm{i}} = {\rm{j}} \cdot {\rm{j}} = {\rm{k}} \cdot {\rm{k}} = 1\) is shown.

Most popular questions for Math Textbooks

A wrench \[30\;{\rm{cm}}\] long lies along the positive\[y\]-axis and grips a bolt at the origin. A force is applied in the direction \[\langle 0,3, - 4\rangle \] at the end of the wrench. Find the magnitude of the force needed to supply \[100\;{\rm{N}} \cdot {\rm{m}}\] of torque to the bolt.

Find \(|u \times v|\) and determine whether \(u \times v\) is directed into the page or out of the page.

Find \(|u \times v|\) and determine whether \(u \times v\) is directed into the page or out of the page.

To find: The volume of the parallelepiped determined by the vectors a, b and c.

Find the resultant vector of \({\rm{k}} \times ({\rm{i}} - 2{\rm{j}})\) using cross product.

Want to see more solutions like these?

Sign up for free to discover our expert answers

Get Started - It’s free

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free