having problems with vectors?

Question

Hey i really need help how to do these types of questions i have an exam coming up any help would be greatly appreciated thanks

http://img295.imageshack.us/img295/4956/vectrsit3.jpg

Answer 1

A vector in R^2 with magnitude r and direction θ can be written in rectangular form as (r cos θ, r sin θ) = (r cos θ) i + (r sin θ) j.
So in this case x = (4 cos π/3, 4 sin π/3) = (4.(1/2), 4.(√3/2)) = (2, 2√3).

a) ||y|| = √(2^2 + (-3)^2) = √13.
||z|| = √(4^2 + 1^2) = √17.

b) x = (2, 2√3), y = (2, -3)
So x . y = 2(2) + (2√3) (-3)
= 4 - 6√3.

c) By drawing a perpendicular you should be able to convince yourself that the area of a triangle with vectors x and y forming two of the sides is ||x×y||/2.
So here we have x = (2, -3, 0) and y = (4, 1, 0) as two of the side vectors (since the origin is one vertex, the vectors from there to the other two vertices match the coordinates of those vertices; the other side vector is ± (2, 4, 0) depending on which way you traverse it).
Now x×y = (-3, 0, 14) and so the area is ||(-3, 0, 14)|| / 2
= √(3^2 + 14^2) / 2
= √205 / 2.