help: cal 3 - the cross product?

Question

i have no idea how to do these:

1. let a = a_1i + a_2j and b = b_1i + b_2j be nonzero vectors in the xy plane. Show that a x b is parallel to k

2. show that (a x b) * b = 0 for all vectors a and b

3. let a, b and c be vectors. which of the following expressings make sense and which do not? explain you answer in each case.

a.) a * (b x c)
b.) a x (b * c)
c.) a * (b * c)
d.) a x (b x c)

Answer 1

1 a and b are vectors in the x y plane because they have no k component. the cross product is a vector perpendicular to a and b so it must be parallel to k, You can also prove it by writing a and b with a zero k component and formally taking the Cross product.

2 axb is perpendicular to b and so dotted with b=0

3 b*c is not a vector but a scalar and a vectorX scalar does not make sense.