Calculating the Dot Product: A*B

[Crauven]

I need some major help with some dot products. I was curious if when getting the dot product, you add the angles of each one to each other. I'll give the problem:

theta=(/)

Find A*B(dot product)
A=8.6i+5j
B=9.7i+2.6j

So i used the formula my teacher gave me to find the angle, theta, tan(/)=y/x. So i got the Angle for each:
tan(/)=5/8.6
(/)=tan^-1(5/8.6)
(/)=30.2

and then for B:
tan(/)=2.6/9.7
(/)=tan^-1(2.6/9.7)
(/)=15

So now, I am given two formulas to get the dot product
A*B=ABcos(/)
AB means the magnitude of A multiplied by the magnitude of B. I can get that, but it asks for one angle... what do I do? Add the two angles together? Or subtract them? I'm confused, please help!

 

[chroot]

In the formula

$$\vec A \cdot \vec B = || \vec A || \, || \vec B || \cos \theta$$

the angle $\theta$ is the angle between the two vectors. When you found the "angle" for each vector, what you were really finding was the angle between each vector and the positive x-axis. If one vector makes an angle of 30.2 degress with the positive x-axis, and the other makes an angle of 15 degrees, the angle between them is 30.2 - 15 = 15.2 degrees.

When it doubt, draw a picture! You'll see that the two vectors make an angle of 15.2 degrees between them.

- Warren

 

[NateTG]

For the lazy among us, there's an alternative formula:
if
$$\vec{A}=<A_x,A_y>$$
and
$$\vec{B}=<B_x,B_y>$$
then
$$\vec{A}\cdot\vec{B}=A_xB_x+A_yB_y$$

 

[Crauven]

I use both formulas, the first that Warren mentioned to get the answer, then i use the lazy way to check it. Thanks guys!