/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q. 57 Show that for any vector \(v\) i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Q. 57 (page 813)

Show that for any vector \(v\) in \(\mathbb{R}^{3}\),

\(v = (v 路 i)i + (v 路 j)j + (v 路 k)k\).

Short Answer

Expert verified

It is shown that for any vector \(v\) in \(\mathbb{R}^{3}\), \(v = (v 路 i)i + (v 路 j)j + (v 路 k)k\).

Step by step solution

01

Step 1. Show

We have to show that \(v = (v 路 i)i + (v 路 j)j + (v 路 k)k\). Now, as we know vector \(v\) is in \(\mathbb{R}^{3}\).

Let \(v=ai+bj+ck\)

So, \(v\cdot i=a\left ( 1 \right )\)

\(v\cdot i=a\) ......(a)

And

\(v\cdot j=b\left ( 1 \right )\)

\(v\cdot j=b\) .......(b)

And

\(v\cdot k=c\left ( 1 \right )\)

\(v\cdot k=c\) ........(c)

02

Step 2. Show

Now, take R.H.S of \(v = (v 路 i)i + (v 路 j)j + (v 路 k)k\) and by using equations (a), (b), and (c) we get,

\((v\cdot i)i + (v\cdot j)j + (v\cdot k)k=ai+bj+ck\)

\((v\cdot i)i + (v\cdot j)j + (v\cdot k)k=v\)

Hence proved.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose f and g are functions such that limx鈫�3f(x)=5,limx鈫�4f(x)=2andlimx鈫�3g(x)=4

Given this information, calcuate the limits that follow, if possible. If it is not possible with the given information, explain why.

limx鈫�3-2f(x)

If u, v, and ware three mutually orthogonal vectors in 鈩�3, explain why u脳(v脳w)=0.

Find u+vandu-valso sketch u,v,u+v,u-v

u=4,0andv=0,3

In Exercises 36鈥�41 use the given sets of points to find:

(a) A nonzero vector N perpendicular to the plane determined by the points.

(b) Two unit vectors perpendicular to the plane determined by the points.

(c) The area of the triangle determined by the points.

P(1,6),Q(0,鈭�3),R(鈭�5,4)

(Hint: Think of the xy-plane as part of 鈩�3.)

Find a unit vector in the direction opposite to3,4,5.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.