Chapter 12: Q. 3 (page 952)

How many unit vectors are there in $ℝ^{1}$ ? How many unit vectors are there in $ℝ^{n}$ for $n > 1$ ?

Short Answer

Expert verified

There are infinite vectors in $R^{2}$ . Similarly, there are 2 unit vectors in $R^{n} .$

Step by step solution

Introduction

One of the unit vector in $□^{1}$ is i

Since $i$ is a unit vector, so

$| i | = 1$

Even,

$| - i | = \sqrt{(- 1)^{2}}$

$= \sqrt{1}$

$= 1$

- i

is also a unit vector in

□^{1}

Hence, there are two unit vectors $i$ and $- i$ in $□^{'}$ .

Calculation of vectors

For $n > 1$

Consider a vector of the form $u = α i + β j$ where $α$ and $β$ are any real number. The vector is in the space, $□^{2}$ i.e. $n = 2 > 1$ .

The unit vector of $u = α i + β j$ is $\frac{α}{\sqrt{α^{2} + β^{2}}} i + \frac{β}{\sqrt{α^{2} + β^{2}}} j$ . Since $α$ and $β$ are any real number, so $α$ and $β$ are infinite numbers.

There are infinite unit vectors in $□^{2}$ . Similarly, there are infinite vectors in $□^{n} .$

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

How many unit vectors are there in $ℝ^{1}$ ? How many unit vectors are there in $ℝ^{n}$ for $n > 1$ ?

Short Answer

Step by step solution

Introduction

Calculation of vectors

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Decision Maths

Discrete Mathematics

Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

How many unit vectors are there in ℝ1 ? How many unit vectors are there in ℝn for n>1 ?

Short Answer

Step by step solution

Introduction

Calculation of vectors

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Decision Maths

Discrete Mathematics

Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

How many unit vectors are there in $ℝ^{1}$ ? How many unit vectors are there in $ℝ^{n}$ for $n > 1$ ?