3D Vectors + Angles

Hi. This question is from an OCR MEI paper- Q8 iii.June 2012.

To find theta, in Part iii, you have to use the direction vectors of A'B and AB and then use the dot product but I'm confused as to why these vectors produce theta? Why wouldn't they make the angle ABA'?

Thank you

A'BC is a straight line, so the lines A'B and BC will have the same direction vectors.

Oh, yes! That makes much more sense now!! Thank you!

Also, sorry to bug you, if I wanted to find the angle ABA' (just using vectors), would I use the direction vectors AB and BA'?

You'd want BA and BA'. Both vectors most come out of the angle, or both must go into the angle; you can't mix one of each.

I understand that but then where I'm actually working out theta, I've used AB and A'B (A'B being equivalent to BC), which goes in and then out of the angle? Does that mean that the mark scheme's wrong or is A'B equivalent to CB?

I'm sorry, I'm actually very confused about the directions!

The vector A'B is (-2,-1,1), and this is a vector in the same direction as BC

The vector BA is (1,-1,2).

And these are the two vectors used in the mark scheme.

Are you sure you used AB? (-1,1,-2)

And then for ABA', I can use BA (goes outwards) and BA' (goes outwards) or AB (goes inwards) and A'B (goes inwards)!!