STUPID Vector qusetion - dont understand dot product rule

[thomas49th]

Homework Statement

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

Homework Equations

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 :)

 

[BobMonahon]

Simple answer: A dot B is a number (scalar) - not a vector. It's definition is: a_xb_x + a_yb_y + a_zb_z.

 

[Mark44]

Homework Statement

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)

You're using row notation. Columns are vertical.

Prove that OA is perpendicular to AB

Are you sure you have written the problem correctly? Vectors OA and OB are perpendicular, but OA and AB aren't.

Homework Equations

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)

Where did you get (3, 3, 3)? Vector AB = OB - OA, which is (1, 1, -4) - (2, 2, 1) = (-1, -1, -5).

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 :)

 

[Unco]

You're using row notation. Columns are vertical.

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

 

[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?"

Here is one answer that might help you make sense of it:
Look at the dot-product of A.A: it is simply the Pythagorean Theorem.
If A = (a_x, a_y, a_z)
Then A.A = a_x² + a_y² + a_z².
So ... the dot-product is a method of determining the length of a vector: Lth (A) = $$\sqrt{A.A}$$

That's just an example to make you feel comfortable with the rule - because it's useful and makes sense in that situation.

Good luck.

BobM