Question

A parallelogram is formed by the vectors = (2, 3) and = (1,

1).
a) Determine the lengths of the diagonals.
b) Determine the perimeter of the parallelogram.

Answer 1

a) 5 and √5

b) P = 2* √13 + 2* √2

Explanation:

We add the two vector and for definition the result will be one of the diagonal of the parallelogram. Then

vector OA ( 2 , 3 ) vector OB ( 1 . 1 )

If vector OD = OA + OB then

coordinates of OD will be ( 2 + 1 , 3 + 1 ) ( 3 , 4 )

And the length of OD is according to Pythagoras Theorem

|OD| = √ (3)² + (4)² =√ 9 + 16 = √25 = 5

For the other diagonal we need to apply the subtraction of vectors wich will give us the other diagonal

vector OA = ( 2 , 3 ) and vector OB = ( 1 , 1 )

If vector BA is the difference between vectors OA - OB then vector BA is

vector BA = ( 2- 1 , 3- 1 ) = ( 1 , 2 )

And the length of BA is according to Pythagoras Theorem

BA = √(1)² + (2)² = √1 + 4 = √5

Then the length of the other diagonal is √ 5

b) To find the perimeter of the parallelogram we need to apply

Perimeter = 2 OA + 2 OB

P = 2 OA + 2 OB (1)

So length of OA is:

|OA| = √(2)² + (3)² = √ 13

and

|OB| = √(1)² + (1)² = √2

Then by subtitution in (1)

P = 2* √13 + 2* √2