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Chapter 12: Q. 3 (page 952)

How many unit vectors are there in 鈩�1 ? How many unit vectors are there in 鈩�n for n>1 ?

Short Answer

Expert verified

There are infinite vectors in R2. Similarly, there are 2 unit vectors in Rn.

Step by step solution

01

Introduction

One of the unit vector in 鈻�1is i

Since iis a unit vector, so

|i|=1

Even,

|-i|=(-1)2

=1

=1

-iis also a unit vector in 鈻�1.

Hence, there are two unit vectors i and -i in 鈻�'.

02

Calculation of vectors

For n>1

Consider a vector of the form u=伪i+尾jwhere 伪and 尾are any real number. The vector is in the space, 鈻�2i.e.n=2>1.

The unit vector of u=伪i+尾jis 伪伪2+尾2i+尾伪2+尾2j. Since 伪and 尾are any real number, so 伪and 尾are infinite numbers.

There are infinite unit vectors in 鈻�2. Similarly, there are infinite vectors in 鈻�n.

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When you use the method of Lagrange multipliers to find the maximum and minimum of f(x,y)=x+ysubject to the constraint xy=1,you obtain two points. Is there a relative maximum at one of the points and a relative minimum at the other? Which is which?

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