Question

Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are vectors in the \(x y\) -plane and a and \(c\) are scalars. $$(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})$$

Short answer

Question: Prove the vector property \((\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})\) using components. Answer: Using components, we find that both operations result in the vector \(\begin{bmatrix} u_1+v_1+w_1 \\ u_2+v_2+w_2 \end{bmatrix}\), proving the vector property \((\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})\). Geometrically, this property can be illustrated by drawing the vectors on the \(xy\)-plane and showing that both operations lead to the same final point.

Step-by-step solution

Represent vectors in terms of components

Let \(\mathbf{u} = \begin{bmatrix} u_1 \\ u_2 \end{bmatrix}\), \(\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}\), and \(\mathbf{w} = \begin{bmatrix} w_1 \\ w_2 \end{bmatrix}\). These are the components of vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) in the \(xy\)-plane.

Perform operations on both sides of the equation

Now, we will perform the operations on both sides of the equation and see if they are equal. On the left side, we have \((\mathbf{u}+\mathbf{v})+\mathbf{w}\): $$\begin{bmatrix} u_1 \\ u_2 \end{bmatrix} + \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} u_1+v_1 \\ u_2+v_2 \end{bmatrix}$$ Then, add vector \(\mathbf{w}\): $$\begin{bmatrix} u_1+v_1 \\ u_2+v_2 \end{bmatrix} + \begin{bmatrix} w_1\\ w_2 \end{bmatrix} = \begin{bmatrix} u_1+v_1+w_1 \\ u_2+v_2+w_2 \end{bmatrix}$$ On the right side, we have \(\mathbf{u}+(\mathbf{v}+\mathbf{w})\): $$\begin{bmatrix} v_1 \\ v_2 \end{bmatrix} + \begin{bmatrix} w_1 \\ w_2 \end{bmatrix} = \begin{bmatrix} v_1+w_1 \\ v_2+w_2 \end{bmatrix}$$ Then, add vector \(\mathbf{u}\): $$\begin{bmatrix} u_1 \\ u_2 \end{bmatrix} + \begin{bmatrix} v_1+w_1 \\ v_2+w_2 \end{bmatrix} = \begin{bmatrix} u_1+v_1+w_1 \\ u_2+v_2+w_2 \end{bmatrix}$$

Compare the results

We can see that both sides result in: $$\begin{bmatrix} u_1+v_1+w_1 \\ u_2+v_2+w_2 \end{bmatrix}$$ This shows that \((\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})\) using components.

Illustrate the property geometrically

First, draw the vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) on the \(xy\)-plane. Then, follow these steps: 1. Draw the vector \(\mathbf{u}\) starting from the origin. 2. Draw the vector \(\mathbf{v}\) with its tail at the head of vector \(\mathbf{u}\), and vector \(\mathbf{w}\) with its tail at the head of vector \(\mathbf{v}\). 3. Repeat step 1, but this time: a. Draw the vector \(\mathbf{v}\) with its tail at the head of vector \(\mathbf{u}\), and vector \(\mathbf{w}\) with its tail at the head of vector \(\mathbf{v}\), creating a new vector, \(\mathbf{v} + \mathbf{w}\). b. Draw the vector \(\mathbf{u}\) with its tail at the head of the new vector, \(\mathbf{v} + \mathbf{w}\). You will notice that both paths lead to the same final point. This illustrates the vector property \((\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})\) geometrically.