In this example, we explore the vector operation of combining multiple vectors, specifically calculating the resultant vector of \(-2\mathbf{w} + \mathbf{u} - \mathbf{v}\). The vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) are oriented either horizontally or vertically, which simplifies the process of sketching and calculating the resultant vector.

To begin, we focus on the operation \(\mathbf{u} - \mathbf{v}\). This requires us to represent \(\mathbf{v}\) as a negative vector, \(-\mathbf{v}\), which points in the opposite direction of \(\mathbf{v}\) while maintaining the same magnitude. By placing the tail of \(-\mathbf{v}\) at the tip of \(\mathbf{u}\), we can find the resultant vector \(\mathbf{u} - \mathbf{v}\) by connecting the initial point of \(\mathbf{u}\) to the terminal point of \(-\mathbf{v}\).

Next, we calculate \(-2\mathbf{w}\). If \(\mathbf{w}\) has a length of 4 units to the left, then \(2\mathbf{w}\) will extend 8 units to the left. To find \(-2\mathbf{w}\), we simply reverse the direction of \(2\mathbf{w}\), resulting in a vector that points 8 units to the right.

Now, to find the final resultant vector \(-2\mathbf{w} + (\mathbf{u} - \mathbf{v})\), we apply the tip-to-tail method again. We take the vector \(-2\mathbf{w}\) and place its tail at the tip of the vector \(\mathbf{u} - \mathbf{v}\). The resultant vector is then drawn from the initial point of \(\mathbf{u}\) to the terminal point of \(-2\mathbf{w}\), providing us with the final solution.

This method of vector addition emphasizes the importance of direction and magnitude, allowing us to visualize and compute complex vector operations effectively.