# 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.

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### 1 Answer

When a parallelogram is formed by two vectors, one diagonal is the vector sum of the sides and another diagonal is the vector difference of the sides.

The sum of the vectors (2,3) and (1, 1) is (2+1, 3 + 1) = (3, 4).

The length of the diagonal is the magnitude of this vector:

`sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5`

The difference of the vectors (2, 3) and (1,1) is (2-1, 3 - 1) = (1, 2).

The magnitude of this vector is

`sqrt(1^2 + 2^2) = sqrt(1 + 4) = sqrt(5)` .

**The lengths of the diagonals are 5 and** `sqrt(5)` .

The perimeter of the parallelogram is the sum of the length of all sides.

The length of two of the sides is the magnitude of the vector (2,3) and the length of the two other sides is the magnitude of the vector (1, 1).

The magnitude of the vector (2, 3) is

`sqrt(2^2 + 3^2) = sqrt(4 + 9) = sqrt(13)`

The magnitude of the vector (1, 1) is `sqrt(1^2 + 1^2) = sqrt(2)` .

**The perimeter of the parallelogram is** `2sqrt(13) + 2sqrt(2)` .