Proving distributivity of Dot/Cross product

[IsaacA]

Using the definitions in equations 1.1 and 1.4, and appropriate diagrams, show that the dot product and cross product are distributive;
(a) when the three vectors are coplanar;
(b) in the general case.

Eq. 1.1) A dot B = ABcosθ

Eq. 1.4) A cross B = ABsinθN

This is exactly how my book puts the formulas.

I know how the definition of the dot product is derived, and that it's distributive over vector addition, but I don't understand why they're asking why the three vectors are coplanar. I don't see where the third vector comes into play. I haven't even tried solving this on the cross product side because I know if I don't conceptually grasp the dot product part of it the cross product will only frustrate me.

Here's my attempt at this proof:

Part A: Stared at it for a while trying to figure it out and eventually gave up.

Part B: Broke out the comfort food. Cried a little.

1.2 Is the cross product associative?

(Vector A cross Vector B) cross vector C (equals?) Vector A cross (Vector B cross Vector C)

I know the cross product isn't associate because the order of the cross product determines the direction of the resultant vector, but I feel like there's more to it.

Thank you all so much for your help!

 

[Ray Vickson]

Using the definitions in equations 1.1 and 1.4, and appropriate diagrams, show that the dot product and cross product are distributive;
(a) when the three vectors are coplanar;
(b) in the general case.

Eq. 1.1) A dot B = ABcosθ

Eq. 1.4) A cross B = ABsinθN

This is exactly how my book puts the formulas.

I know how the definition of the dot product is derived, and that it's distributive over vector addition, but I don't understand why they're asking why the three vectors are coplanar. I don't see where the third vector comes into play. I haven't even tried solving this on the cross product side because I know if I don't conceptually grasp the dot product part of it the cross product will only frustrate me.

Here's my attempt at this proof:

Part A: Stared at it for a while trying to figure it out and eventually gave up.

Part B: Broke out the comfort food. Cried a little.

1.2 Is the cross product associative?

(Vector A cross Vector B) cross vector C (equals?) Vector A cross (Vector B cross Vector C)

I know the cross product isn't associate because the order of the cross product determines the direction of the resultant vector, but I feel like there's more to it.

Thank you all so much for your help!

The question is asking you to prove $\vec{A}\cdot(\vec{B} + \vec{C}) = \vec{A}\cdot \vec{B} + \vec{A} \cdot \vec{C},$ and the same type of result using $\times$ instead of $\cdot$ . So, of course there have to be three vectors involved.

RGV