Question

State whether or not the following statements are true. Justify your reasoning.?

a. Vector a • (Vector b + Vector c) = Vector a • Vector b + Vector a • Vector c

b. Vector a × (Vector b + Vector c) = Vector a × Vector b + Vector a × Vector c

c. Vector a × (Vector b • Vector c) = Vector a × Vector b • Vector a × Vector c

Any help is appreciated.

Answer 1

I will just do two dimensions for demo

a)
A dot (B+C) = AxBx + Ax Cx + AyBy + AyCy
does that equal
AxBx+AyBy + AxCx+AyCy
Yes, it does

b)
A X (B+C) = A X (Bx+Cx)i +(By+Cy)j

i j k
Ax Ay 0
(Bx+Cx) (By+Cy) 0

find determinant
[Ax(By+Cy) - Ay(Bx+Cx) ]k
k direction of course
[ AxBy + AxCy - AyBx - AyCx ]k
is that the same as
A X B + A X C ???
[ AxBy-AyBx +AxCy-AyCx] k
YES, the same

c)
No way !!!
on left is a vector, on right is a scalar. in fact you can not even do the operation on the left which is a vector cross a scalar.

Answer 2

Q: What do you get when you cross a mosquito with a mountain climber?

A: Nothing -- you cannot cross a vector and a scaler.