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. $$\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$$

Short answer

Question: Prove that the vector addition is commutative using components and illustrate it geometrically. Answer: Vector addition is commutative, meaning \(\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}\) for any two vectors \(\mathbf{u}\) and \(\mathbf{v}\). Using components, we showed that \((u_x + v_x, u_y + v_y) = (v_x + u_x, v_y + u_y)\), hence proving the commutative property. Geometrically, we sketched the vectors \(\mathbf{u}\), \(\mathbf{v}\), \(\mathbf{u}+\mathbf{v}\), and \(\mathbf{v}+\mathbf{u}\) and observed that the head of \(\mathbf{u}+\mathbf{v}\) coincides with the head of \(\mathbf{v}+\mathbf{u}\), confirming that vector addition is commutative.

Step-by-step solution

Prove Commutative Property using Components

Firstly, let's express the vectors \(\mathbf{u}\) and \(\mathbf{v}\) in terms of their components: $$\mathbf{u} = (u_x, u_y)$$ $$\mathbf{v} = (v_x, v_y)$$ Now we need to prove that \(\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}\) using their components. Start by adding the components of \(\mathbf{u}\) and \(\mathbf{v}\): $$(u_x, u_y) + (v_x, v_y) = (u_x + v_x, u_y + v_y)$$ Next, add the components of \(\mathbf{v}\) and \(\mathbf{u}\): $$(v_x, v_y) + (u_x, u_y) = (v_x + u_x, v_y + u_y)$$ Since \((u_x + v_x) = (v_x + u_x)\) and \((u_y + v_y) = (v_y + u_y)\), we can conclude that adding the components of \(\mathbf{u}\) and \(\mathbf{v}\) yields the same result as adding the components of \(\mathbf{v}\) and \(\mathbf{u}\): $$(u_x + v_x, u_y + v_y) = (v_x + u_x, v_y + u_y)$$ Therefore, \(\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}\).

Sketch the Vectors Geometrically

To illustrate the commutative property geometrically, let's sketch the vectors \(\mathbf{u}\), \(\mathbf{v}\), and their sum \(\mathbf{u}+\mathbf{v}\). 1. Draw the vector \(\mathbf{u}\) with its tail at the origin and its head at point A, which has coordinates \((u_x, u_y)\). 2. Draw the vector \(\mathbf{v}\) with its tail at the origin and its head at point B, which has coordinates \((v_x, v_y)\). 3. Now draw the vector \(\mathbf{u}+\mathbf{v}\) with its tail at the origin and its head at point C, which has coordinates \((u_x+v_x, u_y+v_y)\). To do this, simply add the x-components of \(\mathbf{u}\) and \(\mathbf{v}\) to find the x-coordinate of point C, and add the y-components of \(\mathbf{u}\) and \(\mathbf{v}\) to find the y-coordinate of point C. 4. Similarly, draw the vector \(\mathbf{v}+\mathbf{u}\) with its tail at the origin and its head at point D, which has coordinates \((v_x+u_x, v_y+u_y)\). 5. Observe that points C and D have the same coordinates, indicating that the head of the vector sum \(\mathbf{u}+\mathbf{v}\) is in the same position as the head of the vector sum \(\mathbf{v}+\mathbf{u}\). This confirms that vector addition is commutative geometrically.