vector question

[chandra21]

a vector is equal to a vector which is parallal with it and has the the same direction and same length as it. Then how can distance be defined by to vector?

 

[daon2]

a vector is equal to a vector which is parallal with it and has the the same direction and same length as it. Then how can distance be defined by to vector?

I'm not sure what you're asking, did you mean length?The normal length of a vector in \(\displaystyle \mathbb{R}^n\) is just the square root of the sum of the squares of its components. In general there are many different kinds of lengths you may define. For example, even in \(\displaystyle \mathbb{R}^2\) you can define the length of \(\displaystyle (x,y)\) as \(\displaystyle \text{max}\{|x|,|y|\}\).

 

[chandra21]

i mean to say that we can move vectors parallal to its support. Then how can we define the parpendicular distance between two vectors?

 

[daon2]

i mean to say that we can move vectors parallal to its support. Then how can we define the parpendicular distance between two vectors?

What do you mean by "move parallel to the support" of a vector? And are we talking about arbitrary vectors, not necessarily in \(\displaystyle \mathbb{R}^n\)? Please explain your question and give any necessary background information that might be relevant in very complete detail.

 

[chandra21]

yes.i am talking about the vector in 3d. It should be known to all that a vecter is equal to anothor vector if the have same or parallal support,same sence and same direction. So how can we fixed a vector in a closed area?

 

[daon2]

yes.i am talking about the vector in 3d.

You never mentioned this. You should explicitly state that you are considering the vector space \(\displaystyle \mathbb{R}^3\).

It should be known to all that a vecter is equal to anothor vector if the have same or parallal support,same sence and same direction.

I'm not sure if you're stating that it should be common knowledge or not, if you are, that's a little silly. You have not defined what "parallel support" is as I asked, either. Now I'll also need you to define "sence."

So how can we fixed a vector in a closed area?

I am sorry, but I still have no idea what you are asking.

 

[chandra21]

what i want you to say will become easier if you tell me what are the condition for equality of two vectors in 3d.

 

[pka]

what i want you to say will become easier if you tell me what are the condition for equality of two vectors in 3d.

In \(\displaystyle \mathbb{R}^3\) a vector is a triple of real numbers.

So, \(\displaystyle <a,b,c>=<p,q,r>\text{ if and only if }a=p,~b=q,~\&~c=r~.\) (Those are all real numbers.)

 

[chandra21]

ok. Vector space apart if we consider vector with direction and think it graphically,that is it has an initial and terminal point then how can you define the condition of equality?

 

[pka]

ok. Vector space apart if we consider vector with direction and think it graphically,that is it has an initial and terminal point then how can you define the condition of equality?

Vectors are not physical objects. Learn that!

Many physical objects, real world ideas, scientific concepts can be represented as a vector. But a vector itself is none of those things. It is a number triple if we are in 3space.

In your example, suppose there are points, \(\displaystyle P: (1,-2,0)~\&~Q: (3,4,-1)\).
We often say the \(\displaystyle <2,6,-1>\) represents the 'action' of 'moving' from \(\displaystyle P\text{ to Q}.\)
We even symbolize it as \(\displaystyle \overrightarrow {PQ} \).

BUT the vector \(\displaystyle <2,6,-1>\) also symbolizes going from point \(\displaystyle A: (-7,2,5)\) to point \(\displaystyle B: (-5,8,4)\).

Now quite clearly \(\displaystyle A\ne P~\&~B\ne Q\), but nevertheless \(\displaystyle \overrightarrow {PQ}=\overrightarrow {AB}=<2,6,-1>\)

Many good textbooks on vectors stress the point that we can think of vectors as having length & direction.
Now they do that even though in the mathematics of 3space, a vector is a number triple.