Where should the origin be for proving midpoints form a parallelogram?

[gracy]

While proving the Midpoints of the Sides of a Quadrilateral Form a Parallelogram , I got bogged down with position vectors.

Let a,b,c and d be the position vectors of A,B,C and D. But where is the origin? Aren't we supposed to locate position of origin?

 

[PeroK]

Let a,b,c and d be the position vectors of A,B,C and D. But where is the origin? Aren't we supposed to locate position of origin?

You can put the origin wherever you like. I might put it at point $A$.

 

[gracy]

If we take origin at A, position vector of A that is given to be a will be 0,0 . Right?

 

[cnh1995]

As PeroK said, you can put the origin at any point as per your convenience.

While proving the Midpoints of the Sides of a Quadrilateral Form a Parallelogram ,

This problem can be solved using simple properties of triangle.

 

[PeroK]

If we take origin at A, position vector of A that is given to be a will be 0,0 . Right?

I'd say the position vector of $A$ in that case is $\vec{0}$. This may simplify the problem.

 

[gracy]

I want to use the following formula for position vector of mid point

For that I need origin other than point A.

 

[PeroK]

I want to use the following formula for position vector of mid point

For that I need origin other than point A.

That's the right formula, but it's even simpler with $\vec{OA} = \vec{0}$.

 

[gracy]

If we take $\vec{OA}$ = $\vec{0}$
The formula will be reduced to
$\vec{OM}$ = $\frac{OB}{2}$
(I meant position vector of OB , I don't know how to get vector sign on top of OB)

 

[PeroK]

If we take $\vec{OA}$ = $\vec{0}$
The formula will be reduced to
$\vec{OM}$ = $\frac{OB}{2}$
(I meant position vector of OB , I don't know how to get vector sign on top of OB)

Okay, that gives you the position vector of point $P$.

Have you thought yet about what you need to do to show that $PQRS$ is a parallelogram?

 

[gracy]

In the book it's given
PQ= position vector of Q - position vector of P
How so? Is there any particular standard formula for this that I am missing?

 

[PeroK]

In the book it's given
PQ= position vector of Q - position vector of P
How so? Is there any particular standard formula for this that I am missing?

It's not a formula. But, what defines a parallelogram?

 

[gracy]

what defines a parallelogram?

A Parallelogram has opposite sides parallel and equal in length.

 

[PeroK]

Parallelogram has opposite sides parallel.

Good. Think a bit more about what you need to do to show this.

In the book it's given
PQ= position vector of Q - position vector of P
How so? Is there any particular standard formula for this that I am missing?

You can get from the origin to point $Q$ in two ways:

$\vec{OQ}$

Or:

$\vec{OP} + \vec{PQ}$

Therefore:

$\vec{OQ} = \vec{OP} + \vec{PQ}$

 

[gracy]

I think I've got it now. Thanks @PeroK @cnh1995