Proof of Cosine law using vectors

[Andrewlorenzo]

Homework Statement

Two vectors of lengths a and b make an angle θ with each other when placed tail to tail. Prove, by taking components along two perpendicular axes, that the length of the resultant vector is r=√(a^2+b^2+2abcos θ )

Homework Equations

Would this be correct?
How would you simplify it?:uhh:

The Attempt at a Solution

So ax = acos θ , bx = bcos θ , ay = asin θ , by = bsin θ. So rx = ax + bx and ry = ay + by . So r=√(r_x+r_y ) .
Which means that, √(〖[(a+b)cosθ]〗^2+〖[(a+b)sinθ]〗^2 )

 

[rl.bhat]

Let vector a be along x-axis and vector b makes an angle theta with x-axis.
Now take the component of b along x and y axis. Find net x component and y component and find the resultant.

 

[nlightner]

=√((a+b)^2 (cos^2θ+sin^2θ))=√((a+b)^2 )=√(a^2+b^2+2abcos θ )
Hello,

Your attempt at a solution is correct. To simplify it, you can use the Pythagorean identity (cos^2 θ + sin^2 θ = 1) to simplify the expression inside the square root to (a + b)^2. This will give you a final result of r = √(a^2 + b^2 + 2abcos θ).