Question

the three vertices of a parallelogram are(p+q,p-q);(2p+q,2p-q),(p-q,p+q).find the fourth vertex.

Answer 1

Answer:

A(a + b, a - b), B(2a+b, 2a-b), C(a - b, a + b)

Then:

Origin: O(0, 0)

That is uppercase letter O and the coordinates are zero and zero.

If you draw it, you can see the fact that:

D ———————— C

| ……………………….. |

| ……………………….. |

A ———————— B

Vector OD = OA + BC

Also, without drawing, the definition of a parallelogram means that vector AB is the same (direction and modulus) as vector DC and, also, that vector BC is the same as vector AD.

So, adding vector BC to OA is the same as adding AD to OA.

Since “OA” means “from O to A” …

OA + AD means “from O to A and then from A to D” … which is the same as the vector “from O to D”, which is exactly the vector OD.

And, what is BC if you know B (the same as OD) and C (the same as OC) ??

Well:

BC = C-B

This means: “from BC” = “from B to O and then from O to C”

But “vector from B to O” is the opposite as “from O to B”, which is OB = B

BC = BO + OC = OC - OB = C-B

You don’r need any middle point of one diagonal, it’s just addition and substraction:

D = A + BC = A + C - B

(a + b, a - b) - (2a+b, 2a-b) + (a - b, a + b) =

= ( a + b - [2a+b] + a-b, a-b - [2a-b] + a+b) =

= (- b, b)