Question

Two sides of a parallelogram are formed by the vectors \(\mathbf{u}\) and \(\mathbf{v}\). Prove that the diagonals of the parallelogram are \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{u}-\mathbf{v}\)

Short answer

Answer: The diagonals of the parallelogram are \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{v}-\mathbf{u}\).

Step-by-step solution

Recall the properties of parallelograms

To solve this problem, you should recall that in a parallelogram, opposite sides are parallel and equal in length. Therefore, as the sides of the parallelogram are determined by the given vectors \(\mathbf{u}\) and \(\mathbf{v}\), the opposite sides are also represented by the same vectors.

Define the vertices of the parallelogram using vectors

Let the vertices of the parallelogram be \(A\), \(B\), \(C\), and \(D\). Without loss of generality, let \(\mathbf{u}\) be the vector from vertex \(A\) to vertex \(B\) and let \(\mathbf{v}\) be the vector from vertex \(A\) to vertex \(C\). Then the opposite vertices are connected by the vectors \(\mathbf{AB}=\mathbf{u}\), \(\mathbf{BC}=\mathbf{v}\), \(\mathbf{CD}=\mathbf{u}\), and \(\mathbf{DA}=\mathbf{v}\).

Compute the diagonals using vector addition

The diagonals of the parallelogram can be now computed using vector addition. The diagonal \(\mathbf{AC}\) connects vertices \(A\) and \(C\). To find this diagonal, we start at vertex \(A\), follow the vector \(\mathbf{u}\) to get to vertex \(B\), and then follow the vector \(\mathbf{v}\) from vertex \(B\) to vertex \(C\). In terms of vector addition, this can be written as \(\mathbf{AC}=\mathbf{u}+\mathbf{v}\). Similarly, the diagonal \(\mathbf{BD}\) connects vertices \(B\) and \(D\). To find this diagonal, we start at vertex \(B\), follow the vector \(\mathbf{u}\) in the opposite direction (which gives us \(-\mathbf{u}\)) to get to vertex \(D\), and then follow the vector \(\mathbf{v}\) from vertex \(D\) to vertex \(B\). In terms of vector addition, this can be written as \(\mathbf{BD}=-\mathbf{u}+\mathbf{v}\) which is equivalent to \(\mathbf{BD}=\mathbf{v}-\mathbf{u}\).

Conclusion

Based on our computations, we have shown that the diagonals of the parallelogram are \(\mathbf{AC}=\mathbf{u}+\mathbf{v}\) and \(\mathbf{BD}=\mathbf{v}-\mathbf{u}\). Since this was the desired result, the proof is complete.