Confused about unit vector equivalence

In summary, the textbook states that when x is a unit vector in \Re^{2}, dividing it by its length ||x|| results in \frac{1}{||x||}x. This is because the vector must be shrunk or stretched to have a length of 1, and the reciprocal of the length serves as the factor for this transformation. This concept is similar to the scalar version, where dividing by a number results in a value of 1.
  • #1
magisbladius
7
0
x is a unit vector [tex]\in \Re^{2}[/tex]. My textbook states that [tex]\frac{x}{||x||}=\frac{1}{||x||}x[/tex]. What is the point of including [tex]\frac{1}{||x||}[/tex]; why do they divide the vector by its length?

Edit: I just looked at a book in Google's database, and from what I understand:

e.g. [tex]\sqrt{{2^2+2^2+1^2}}=3[/tex] so that becomes [tex](\frac{2}{3}) ,(\frac{2}{3}),(\frac{1}{3}) = 1[/tex] due to the the vector rule (add by component). Basically, the answer to my question lies in the proof.
 
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  • #2
We shrink or stretch the vector such that it points the same direction with the original vector but its length is 1. That shrinking or stretching factor is the reciprocal of its length. So, basically we don't do anything different than the scalar version [tex]5\frac{1}{5} = 1[/tex], only difference is that the length notion becomes [tex]\sqrt{a_1^2 + a_2^2 + \ldots + a_n^2}[/tex].
 

Related to Confused about unit vector equivalence

What is a unit vector?

A unit vector is a vector with a magnitude of 1 and is used to represent a direction in space. It is often denoted by a hat symbol (^) above the vector's variable name.

Why is unit vector equivalence confusing?

Unit vector equivalence can be confusing because it involves understanding the concept of vector components and how they relate to one another. It also requires knowledge of vector operations and how they affect the magnitude and direction of a vector.

How do I know if two unit vectors are equivalent?

Two unit vectors are equivalent if they have the same magnitude and direction. This means that they represent the same direction in space, even if they are expressed in different coordinate systems or have different variable names.

Can unit vectors be added or subtracted?

Yes, unit vectors can be added or subtracted using vector addition and subtraction rules. The resulting vector will still have a magnitude of 1 and will represent the combined direction of the original unit vectors.

How are unit vectors used in science?

Unit vectors are commonly used in science, especially in physics and engineering, to represent and analyze the direction and magnitude of quantities such as velocity, force, and acceleration. They are also used in vector calculus to solve mathematical problems involving vectors.

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