Vector Proofs using vector components

[lizette]

Hi! I'm new to the forums. I'm taking an introduction to physics class this semester and I've been having some difficulty with it. Oh, I also wanted to let you know that it's been a while since I've taken calculus or any other math class for that matter. But I need physics to graduate. Anywho ... the question that I have deals with vector components.

Two vectors of magnitudes a and b make an angle theta (which I'll represent as @) with each other when placed tail to tail. Prove, by taking components along two perpendicular anes, that

r = the square root of (a^2 + b^2 + 2abcos@)

gives the magnitude of the sum vector R (vector R = r with that arrow above it) of the two vectors.

Well this is what I have so far:

vector A = Axi + Ayi
vector B = Bxi + Byi
vector R = vector A + vector B

A^2 = Ax^2 + Ay^2
B^2 = Bx^2 + By^2
R^2 = A^2 + B^2

A dot B = A*B = ABcos@

I can see how r = square root of (A^2 + B^2) but where does the 2ABcos@ come in. I have a feeling that it deals with the A*B product, but I don't know how to fit it in.

 

[bogdan]

A dot A=a^2 (a is magn A) A- vector
B dot B=b^2 -the same-
=>
(A+B) dot (A+B)=a^2+b^2+2*A dot B=r^2 (evidently)
...so r^2 =a^2+b^2+2*a*b*cos@...
...do you really have to use components ?

 

[lizette]

well, the problem said to use components .. so i believe so.

thanks for your help!