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 can be expressed as follows: 1. Diagonal AC: \(\mathbf{u}+\mathbf{v}\) 2. Diagonal BD: \(\mathbf{u}-\mathbf{v}\)

Step-by-step solution

Identify properties of a parallelogram

In a parallelogram, opposite sides are parallel and equal in length. Let's denote the vertices of the parallelogram as A, B, C, and D, with AB formed by vector \(\mathbf{u}\), BC formed by vector \(\mathbf{v}\), CD formed by vector \(\mathbf{u}\), and DA formed by vector \(\mathbf{v}\). We will use these properties to find expressions for the vectors representing the diagonals.

Find expression for diagonal AC

To find the vector representing diagonal AC, we can go from point A to point C along two different paths: AB + BC, or AD + DC. Since both paths connect the same two points, their vectors should be the same, so we can write: $$\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AD} + \overrightarrow{DC}$$ Now, substitute the given vectors \(\mathbf{u}\) and \(\mathbf{v}\): $$\overrightarrow{AC} = \mathbf{u} + \mathbf{v}$$ So the vector representing diagonal AC can be expressed as \(\mathbf{u}+\mathbf{v}\).

Find expression for diagonal BD

Similarly, to find the vector representing diagonal BD, we can go from point B to point D along two different paths: BC + CD, or BA + AD. Since both paths connect the same two points, their vectors should be the same, so we can write: $$\overrightarrow{BD} = \overrightarrow{BC} + \overrightarrow{CD} = \overrightarrow{BA} + \overrightarrow{AD}$$ Now, substitute the given vectors \(\mathbf{u}\) and \(\mathbf{v}\) and keep in mind that \(\overrightarrow{BA} = -\mathbf{u}\): $$\overrightarrow{BD} = \mathbf{v} + \mathbf{u} = -\mathbf{u} + \mathbf{v}$$ Now, rewrite the sum to show the diagonal BD in terms of \(\mathbf{u}-\mathbf{v}\): $$\overrightarrow{BD} = \mathbf{u} - \mathbf{v}$$ So the vector representing diagonal BD can be expressed as \(\mathbf{u}-\mathbf{v}\).

Conclusion

In conclusion, given a parallelogram with sides formed by vectors \(\mathbf{u}\) and \(\mathbf{v}\), we have shown that its diagonals can be expressed in terms of these vectors as: 1. Diagonal AC: \(\mathbf{u}+\mathbf{v}\) 2. Diagonal BD: \(\mathbf{u}-\mathbf{v}\)