91Ó°ÊÓ

When two vectors, say \(\mathbf{u}\) and \(\mathbf{v}\), are added, the process is known as the parallelogram rule. Geometrically, it means we create a parallelogram with \(\mathbf{u}\) and \(\mathbf{v}\) as adjacent sides, and the diagonal pointing from their shared initial point to the opposite vertex of the parallelogram represents the resultant vector \(\mathbf{u}+\mathbf{v}\). Similarly, when we subtract one vector from another, say, \(\mathbf{u}-\mathbf{v}\), geometrically it means we need to add \(\mathbf{u}\) and \(-\mathbf{v}\). It's important to note that \(-\mathbf{v}\) has the same magnitude as \(\mathbf{v}\) but opposite direction. When adding \(\mathbf{u}\) and \(-\mathbf{v}\), we apply the parallelogram rule again, forming a parallelogram with \(\mathbf{u}\) and \(-\mathbf{v}\) as adjacent sides.

The main goal is to prove that the magnitude of the difference between two vectors, \(|\mathbf{u}-\mathbf{v}|\), is equal to the distance between the endpoints of \(\mathbf{u}\) and \(\mathbf{v}\). Suppose the vectors are \(\mathbf{u}\ and\ \mathbf{v}\), which share a common initial point. Now consider a parallelogram created using \(\mathbf{u}\) and \(\mathbf{v}\) as adjacent sides. According to the geometric interpretation of vector subtraction, the difference \(\mathbf{u}-\mathbf{v}\) is the vector connecting the endpoint of \(\mathbf{v}\) to the endpoint of \(\mathbf{u}\). This is because we add \(\mathbf{u}\) with the opposite vector \(-\mathbf{v}\). This means the diagonal of the parallelogram formed by \(\mathbf{u}\) and \(\mathbf{v}\) also represents \(\mathbf{u}-\mathbf{v}\). Now we want to find the distance between the endpoints of \(\mathbf{u}\) and \(\mathbf{v}\), which is the length of the diagonal that represents \(\mathbf{u} - \mathbf{v}\). Since the magnitude of a vector is its length, the distance between the endpoints of \(\mathbf{u}\) and \(\mathbf{v}\) is \(|\mathbf{u} - \mathbf{v}|\). Hence, we have shown that \(|\mathbf{u}-\mathbf{v}|\) equals the distance between the endpoint of \(\mathbf{u}\) and the endpoint of \(\mathbf{v}\).