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Unit vectors


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#1 1123581321

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Posted 3 March 2011 - 01:39 AM

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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#2 Schrödinger's hat

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Posted 3 March 2011 - 01:49 AM

We can make all of our equations without unit vectors (but not without vectors), just divide and multiply by the magnitude.
\frac{r}{r}\hat{\mathbf{r}}=\frac{1}{r}\mathbf{r}
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
\frac{1}{r^2} The maginutde (a scalar or number, it carries the units, too) \times\hat{\mathbf{r}} 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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#3 Cap'n Refsmmat

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Posted 3 March 2011 - 01:50 AM

It means vector's length is 1. If you know the dot product, it means that \sqrt{\vec{v} \cdot \vec{v}} = 1. 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:

\vec{F} = G \frac{m_1 m_2}{r^2} \hat{r}

where \hat{r} is a unit vector pointing in the same direction as \vec{r}. In this equation, \vec{r} would be a vector pointing from one object of mass m_1 to another object of mass m_2, and \vec{F} 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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#4 1123581321

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Posted 3 March 2011 - 10:48 AM

In physics were using i, j & k (with hats on them) as unit vectors. Are they used across the board ?
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#5 D H

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Posted 3 March 2011 - 03:41 PM

Across the board? What do you mean by that?

Those certainly are not the only notations used, even for three space. You will see \hat x, \hat y, \hat z to denote the unit vectors along the x, y, and z axes, \hat r, \hat {\theta}, \hat{\phi} for the spherical unit vectors, and more generically \hat u_1, \hat u_2, \hat u_3 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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