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Unit vectors

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I was wondering why we use unit vectors, what's happens if we don't and in the definition of a unit vector, does a magnitude of 1 mean a scalar of 1 (for a that vector) or that the vector itself only represents one/single arrow ?

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We can make all of our equations without unit vectors (but not without vectors), just divide and multiply by the magnitude.

[math]\frac{r}{r}\hat{\mathbf{r}}=\frac{1}{r}\mathbf{r}[/math]

In terms of the magnitude of a unit vector (or any vector) it relates to size.

 

A vector is something that carries two pieces of information, direction and size. So 3km east is a vector, or 1 Newton down. We often use unit vectors when we are supplying the size information elsewhere so we'll have

[math]\frac{1}{r^2}[/math] The maginutde (a scalar or number, it carries the units, too) [math]\times\hat{\mathbf{r}}[/math] the direction (in english this means 'in the direction that you measured r') It still carries a magnitude (1), but this is replaced by the other magnitude when they are multiplied (1x something=something).

 

Another unit vector would be 1 unit east, or 'in the direction I drove'

so you could have 5 km (magnitude) 'in the direction I drove'(unit vector), together they make a normal vector.

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It means vector's length is 1. If you know the dot product, it means that [math]\sqrt{\vec{v} \cdot \vec{v}} = 1[/math]. If you don't, it means that if you take a ruler and measure the vector, it'll be 1 unit long.

 

It's convenient because you can do math with unit vectors without changing the length of your original vector. For example, in physics we might do:

 

[math]\vec{F} = G \frac{m_1 m_2}{r^2} \hat{r}[/math]

 

where [math]\hat{r}[/math] is a unit vector pointing in the same direction as [math]\vec{r}[/math]. In this equation, [math]\vec{r}[/math] would be a vector pointing from one object of mass [math]m_1[/math] to another object of mass [math]m_2[/math], and [math]\vec{F}[/math] would be the gravitational force between them.

 

If we don't use a unit vector in the equation, we end up multiplying by a vector of some random length, and we make the gravitational force vector too strong or too weak.

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Across the board? What do you mean by that?

 

Those certainly are not the only notations used, even for three space. You will see [math]\hat x, \hat y, \hat z[/math] to denote the unit vectors along the x, y, and z axes, [math]\hat r, \hat {\theta}, \hat{\phi}[/math] for the spherical unit vectors, and more generically [math]\hat u_1, \hat u_2, \hat u_3[/math] to denote an arbitrary set of unit vectors. That's just a starter.

 

And of course i, j, and k only pertain to three space.

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