Question

Consider the following statements and choose the correct option:

Statement 1: If `veca` and `vecb` represent two adjacent sides of a parallelogram, then the diagonals are represented by `veca + vecb` and `veca - vecb`.

Statement 2: If `veca` and `vecb` represent two diagonals of a parallelogram, then the adjacent sides are represented by `2(veca + vecb)` and `2(veca - vecb)`.

Which of the following is correct?

Options

• Statement 1 is true and statement 2 is false.

• Statement 2 is true and statement 1 is false.

• Both the statements are true.

• Both the statements are false.

Answer 1

Statement 1 is true and statement 2 is false.

Explanation:

Statement 1: If `veca` and `vecb` represent two adjacent sides of a parallelogram starting from a common vertex, then by the triangle law of vector addition:

One diagonal, which is the resultant vector across the parallelogram, is indeed represented by the sum of the adjacent sides: `veca + vecb`.

The other diagonal, which goes between the tips of the two vectors, is represented by the difference of the adjacent sides: `veca - vecb` (or `vecb - veca`).

The magnitude remains the same.

Thus, statement 1 is correct.

Statement 2: The relationship between diagonals `(vecd_1, vecd_2)` and adjacent sides `(veca, vecb)` is:

`vecd_1 = veca + vecb`

`vecd_2 = veca - vecb` (or `vecb - veca)`

If we are given the diagonals and want to find the sides, we need to solve for `veca` and `vecb`:

Adding the two equations:

`vecd_1 + vecd_2 = (veca + vecb) + (veca - vecb)`

`vecd_1 + vecd_2 = 2veca`

So `veca = 1/2 (vecd_1 + vecd_2)`

Subtracting the second equation from the first:

`vecd_1 - vecd_2 = (veca + vecb) - (veca - vecb)`

`vecd_1 - vecd_2 = 2vecb`

So `vecb = 1/2 (vecd_1 - vecd_2)`

Statement 2 claims the adjacent sides are `2(vecd_1 + vecd_2)` and `2(vecd_1 - vecd_2)`, which is incorrect by a factor of `1/2`.

Thus, statement 2 is false.