# Polar Coordinate Inner Product

## Main Question or Discussion Point

Just starting up school again and having trouble remembering some mathematics. Here's the problem.

Find the inner product of ⃗a = (1, 45◦) and ⃗b = (2, 90◦), where these vectors are in polar coordinates (r, θ).

Thanks =) 1st post here btw.

Related Linear and Abstract Algebra News on Phys.org
http://www.iancgbell.clara.net/maths/vectors.htm

About midway down this page you can see that the dot product in polar coordinates is $\small (r_1,\theta_1) \tiny \bullet \small (r_2,\theta_2) = r_1r_2 \cos(\theta_1-\theta_2)$. One solution is to use this formula.

The other one is just to represent the vectors in normal cartesian coordinates as $\frac{1}{\sqrt{2}}(1,1)$ (the factor 1/sqrt(2) is there to make the vector have length 1) and $(2,0)$ and then use the normal dot product.

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HallsofIvy
1) The "non-coordinate" definition of the dot product: $u\cdot v= |u||v|cos(\theta)$, where $\theta$ is the angle between the two vectors. Here, |u|= 1, |v|= 2 and angle between them is 45 degrees.
2) Convert to Cartesian coordinates. The vector at 45 degrees with length 1 is $<\sqrt{2}/2, \sqrt{2}/2>$. The vector at 90 degrees, with length 2, is <0, 1>.