Scalar Product/Norm proof
Hello, I am having trouble. I would appreciate if someone could help me with part b of this problem, I have solved part a.
Let u = (a,b) and v = (c, d) be nonzero points in the plane and let be the radian measure of the angle with vertex 0 formed by 0, u, and v.
**notations: ||u|| is the norm and <u,v> is the scalar/dot product
a. Prove that
(1) LHS = by definition of dot product
(2) LHS = by distribution and the property that
(3) LHS = by factoring
(4) LHS = RHS by sine cosine rule that
b. I am given that
I have to use this to verify that |ad-bc|/2 is the area of the triangle with vertices 0, u, and v and that that as a consequence |ad-bc| is the area of the parallelogram with vertices 0, u, u+v, and v.
I know that the area of a triangle is , so if the area of the triangle with vertices 0, u, v. Then shouldn't the area be |ac-bd|/2 instead of |ad-bc|/2. Any help is greatly appreciated. Thank you.