Q. 3 - Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1 Introduction

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

Since $i$ is a unit vector, so

                              $\left|i\right|=1$

Even,

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

                                  $=\sqrt{1}$

                                     

                                       $=1$

$-\mathbf{i}$ is also a unit vector in ${\square }^1$.

Hence, there are two unit vectors $\mathrm{i}$ and $-\mathrm{i}$ in ${\square }^{\text{'}}$.

Step 2 Calculation of vectors

For $n>1$

Consider a vector of the form $\mathbf{u}=\alpha \mathbf{i}+\beta \mathrm{j}$ where $\alpha$ and $\beta$ are any real number. The vector is in the space, ${\square }^2$ i.e. $n=2>1$.

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

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