Question

Use vectors to prove that a parallelogram is a rectangle if and only if its diagonals are equal in length.

Short answer

To prove a parallelogram is a rectangle if its diagonals are equal, vectors can be used to represent the sides and diagonals. Equating the squared magnitudes gives a relation, which implies that the vectors representing sides are perpendicular to each other, thus proving that the parallelogram is a rectangle.

Step-by-step solution

Represent the Parallelogram

Represent the parallelogram ABCD using vectors. We will consider one point (for instance, A) as the origin. Now let's denote AB as vector \( \vec{a} \), AD as vector \( \vec{b} \) and AC and BD as vectors \( \vec{d1} \) and \( \vec{d2} \) respectively.

Expression for Diagonals

First, we aim to express the vectors for our diagonals AC and BD i.e., \( \vec{d1} \) and \( \vec{d2} \) in term of vectors \( \vec{a} \) and \( \vec{b} \). The diagonal AC can be given as the sum of vectors AB and BC. Remember that BC = AD = \( \vec{b} \). Hence, \( \vec{d1} = \vec{a} + \vec{b} \). Similarly the diagonal BD can be represented as the sum of vectors BA and AD. Since BA= - AB = -\( \vec{a} \), Hence, \( \vec{d2} = -\vec{a} + \vec{b} \).

Equalize the Diagonals

Now, if the parallelogram is a rectangle, then the lengths of diagonals will be equal. According to the property of vectors, the magnitude of vectors \( \vec{d1} \) and \( \vec{d2} \) will be equal, if these vectors represent equal lengths. So, we equate the magnitudes. We square the magnitudes to eliminate the square root for convenience: \( \vec{d1} \cdot \vec{d1} = \vec{d2} \cdot \vec{d2} \).

Simplify

We will simplify this equation by distributing and simplifying: \( (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = (-\vec{a} + \vec{b}) \cdot (-\vec{a} + \vec{b}) \). This will simplify to \( \vec{a} \cdot \vec{a} + 2\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} = \vec{a} \cdot \vec{a} - 2\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} \). This simplifies to \( 4\vec{a} \cdot \vec{b} = 0 \). If \( \vec{a} \) and \( \vec{b} \) are not equal to zero, then \( \vec{a} \cdot \vec{b} = 0 \). Their dot product is zero, which means they are orthogonal (perpendicular) vectors.

Final Proof

Given that AD is perpendicular to AB (as shown in previous step), and we know for a parallelogram, opposite sides are parallel and equal, hence the shape ABCD is a rectangle. Therefore we have shown in the problem that a parallelogram is indeed a rectangle if its diagonals are equal.