Bob

]]>A vector space could just be all the points in plane. We say that is a two dimensional space.

It could be three dimensional, it which case you would need vectors with three components in the x, y and z directions.

It could also have four, five, .... n dimensions, but you probably won't need to worry about those.

I kept it simple in my example by sticking to two dimensions.

Hope that explains that part. Next question?

Bob

]]>space mean distance or other space where planet exist]]>

Bob

]]>I want to help but I need you to say more than ??? What don't you understand? Or give an example of what you do understand.

Bob

]]>I am not sure what you are asking. So I will go back to basics and explain why vectors and coordinates are part of the same topic.

Maths theorists usually start with vector spaces. To keep it simple, let's make a 2 dimensional vector space.

The vectors **a** and **b** here are not parallel so you may use them as a **basis** for the space. That means that any other vector can be written as a linear combination of **a** and **b**.

To demonstrate this, I've drawn another vector **c**. It could be any vector so I just choose one at random.

I've then made two dotted lines. The first goes through O and A and extends as far as necessary in both directions. The second is parallel to OB but goes through C.

Because OA and OB are not parallel, the dotted lines must cross somewhere. I have called that point P. If you count squares you will see that

You can use this method to make any vector a linear combination of **a** and **b**.

**a** and **b** are said to form a basis for the space. Nothing special about **a** and **b** either. I just drew any two non parallel lines through (0,0).

One special basis is the one made by the unit vectors

and

If they are used as the basis then

And then we define the coordinates of C to be those two numbers and write C = (-6,6)

Bob

]]>i is the unit vector in the x direction and j is the unit vector in the y direction. So when you describe a point as (2,3) what you are really saying is that, starting from the origin, you need to go 2 in the x direction (or 2i) and 3 in the y direction (or 3j) to reach P.

Bob

ps. My computer keeps changing i to I. This is because I have got certain auto-adjust spelling corrections set up. If you go back to the I and change it back to i the auto-adjust usually gives up.

Bob

]]>You have OP = 2i + 3j and OQ = -4i + 7j .

PQ = PO + OQ = -2i - 3j -4i + 7j = -6i + 4j

If you fraw a coordinate diagram and plot the points you'll see what 'i' and 'j' is needed to travel from P to Q.

Bob

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