Units in dot product

[WikiGuru]

When you take the dot product of two vectors both with units meters, is the resulting unit meters^2 or is it meters?

[simplethinker]

It would be m²

A.B (let's pretend that little "." is bigger and centered vertically ) = |A||B|cos(θ) where θ=angle between them. Now, cos(θ) is a dimensionless quantity and both |A| and |B| have dimensions m, so the dimensions are m²

[WikiGuru]

Oh, duh. That's how work is computed (Nm). Haha, brain fart.

edit:
Second question:
What is the meaning of dotting two vectors with units meters? I know that the dot product determines the tendency of two vectors to point in the same direction, but with m² units, it doesn't really make sense.

[Galandros]

I though of that and I got nowhere. And I have to control my curiosity in class...

In our class we give no unit.

[simplethinker]

The dot product of two vectors with both units in meters (or any other unit) can be used as a scaling factor for certain situations (I can't think of a better way to phrase this). For example, if you had the vector r=(r_x meters, r_y) and you wanted to determine it's component in the direction of u=(u_x, u_y) meters (instead of in the directions (1, 0) or (0, 1)), doing r.u and dividing that by |u|² (which is 1m²) gives the component in the direction of u (I think this is called the projection).

This kind of thing appears frequently in Linear Algebra, fluid mechanics, optics and a bunch of other things (those were the first to come to mind).