How Do You Find the Inner Product of Vectors in Polar Coordinates?

[spaderdabomb]

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.

 

[Square]

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.

 

[HallsofIvy]

This is more calculus than "linear algebra". There are two ways to go:
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 &lt;\sqrt{2}/2, \sqrt{2}/2&gt;. The vector at 90 degrees, with length 2, is <0, 1>.