1. ## Basic vector help

Hey MHF.

I'm learning vectors now and I have 2 questions.

How come this is true:

To me, this is not very intuitive. If I have a triangle, the sides do not add up to become the 3rd side as displayed but I'm guessing there is some other factor at play to make this a logical conclusion.

My other question is if there are any videos on khan academy or any sites like that covering this concept? I have searched on Khan but to no avail...He starts discussing vectors in linear algebra but I'm pretty sure this is taught before linear algebra.

Thanks!

2. ## Re: Basic vector help

Paze
It seems to me that you confuse the vectors with the Algebra of segments...
Consider a triangle ABC. then vector AB+vector BC = vector AC ( the third side....
However if AB,AC,BC are segments we have the triangle inequality (length of any side )< summ of the lengths of the two other sides) ..
Please revise the definition of the vectors .

3. ## Re: Basic vector help

Originally Posted by Paze
How come this is true:
To me, this is not very intuitive. If I have a triangle, the sides do not add up to become the 3rd side as displayed but I'm guessing there is some other factor at play to make this a logical conclusion.

Do you understand that a vector is an equivalence class of 'objects' that have a given length and a given direction?

Look at the vector $\vec{a}$. It has those two properties.

If you start at point $B$ and 'put' two $\vec{a}$ you are at point $K$.
Then 'move off as one $\vec{b}$ you end up at point $M$

Thus $2\vec{a}+\vec{b}$ is the vector $\vec{BM}$

Now, there is no real triangle that at all. You see the combination of two vectors is a vector.

4. ## Re: Basic vector help

I do understand that they have a given length and a given direction. Is there a simple way to envision this? Like maybe...

Is this also true for my example/picture if I make some assumptions, namely that segment (can I call it segment?) BK is 30° and segment BM is 45°. BK is 2 kilometers, KM is 1.5 kilometers and BM is thus 3.5 kilometers?

A car starts driving 30° north-east from point B. It drives for 2 kilometers and then takes a sharp turn east and drives for 1.5 kilometers.

The car has driven equally to 3.5 kilometers 45° north-east from its starting point (point B).

5. ## Re: Basic vector help

Originally Posted by Paze
I do understand that they have a given length and a given direction. Is there a simple way to envision this? Like maybe...Is this also true for my example/picture if I make some assumptions, namely that segment (can I call it segment?) BK is 30° and segment BM is 45°. BK is 2 kilometers, KM is 1.5 kilometers and BM is thus 3.5 kilometers?
A car starts driving 30° north-east from point B. It drives for 2 kilometers and then takes a sharp turn east and drives for 1.5 kilometers. The car has driven equally to 3.5 kilometers 45° north-east from its starting point (point B).
You have a too literal view of vectors.
They may represent physical objects but vectors are not physical objects.

Here is a good textbook: A vector Space Approach to Geomentry by Melvin Hausner.
It is a Dover reprint, so not expensive.

6. ## Re: Basic vector help

Thank you. I'll have a look at that!