STUPID Vector qusetion - dont understand dot product rule

1. Jan 18, 2009

thomas49th

1. The problem statement, all variables and given/known data

The points A and B have position vectors a = (2,2,1) and b (1,1,-4) respectively relative to an origin O. (im using column notation for shorthand)

Prove that OA is perpendicular to AB

2. Relevant equations

3. The attempt at a solution

To be perpendicular the angle between the lines is 90

using the dot product rule:

vectors a and b should multiply to give cos 90 (which is 0)

AB = (3,3,-3)-(2,2,1) = (1,1,-4)
so:
(2,2,1).(1,1,-4) = (2,2,-4)

NOW WHAT I DONT UNDERSTAND IS why you add the components i,j,k to get 0. I can see 2 + 2 -4 =0, but why do you do this? Why can you do this

Thanks :)

2. Jan 18, 2009

BobMonahon

Simple answer: A dot B is a number (scalar) - not a vector. It's definition is: axbx + ayby + azbz.

3. Jan 18, 2009

Staff: Mentor

You're using row notation. Columns are vertical.
Are you sure you have written the problem correctly? Vectors OA and OB are perpendicular, but OA and AB aren't.
Where did you get (3, 3, 3)? Vector AB = OB - OA, which is (1, 1, -4) - (2, 2, 1) = (-1, -1, -5).

4. Jan 18, 2009

Unco

It's just ordered-set notation; neither row nor column vectors have commas!

5. Jan 19, 2009

BobMonahon

Hi,
You did show that OA is perpendicular to OB. That's good.

I think your question is "Why does the dot-product rule work? Why do you multiply similar components, and then add up the sum?"

So ... the dot-product is a method of determining the length of a vector: Lth (A) = $$\sqrt{A.A}$$