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. $$a(c \mathbf{v})=(a c) \mathbf{v}$$

Short answer

Short answer: The vector property \(a(c \mathbf{v})=(a c) \mathbf{v}\) can be proven by representing the vectors in component form and comparing the components after performing scalar multiplication. Geometrically, this property is shown by illustrating that multiplying a vector by two scalars in succession and multiplying a vector by the product of those scalars both result in vectors with the same length and direction.

Step-by-step solution

Write the given property in component form

First, we need to represent the given property in terms of the components of the vectors. Let \(\mathbf{v} = (v_x, v_y)\). The vector property can be written as: $$a(c \mathbf{v})=(a c) \mathbf{v}$$ $$a(c (v_x, v_y))=(a c) (v_x, v_y)$$

Perform scalar multiplication on both sides

On the left side, we must multiply a scalar by a vector that has been multiplied by a scalar. On the right side, we multiply the product of two scalars with a vector. Perform these scalar multiplications: $$a(c v_x, c v_y)=(a c v_x, a c v_y)$$

Compare the components

Now, we must compare the components to see if they are equal: Left side vector components: \((acv_x, acv_y)\) Right side vector components: \((acv_x, acv_y)\) Since the components of the vectors on both sides are equal, the given property is proven to be true: $$a(c \mathbf{v})=(a c) \mathbf{v}$$

Geometric illustration

To illustrate this property geometrically, let's draw the original vector \(\mathbf{v}\), the intermediate vector \(c \mathbf{v}\), and the final vectors \(a(c \mathbf{v})\) and \((a c) \mathbf{v}\): 1. Draw vector \(\mathbf{v}\) with components \((v_x, v_y)\). 2. Multiply \(\mathbf{v}\) by scalar \(c\), resulting in vector \(c \mathbf{v}\) with components \((c v_x, c v_y)\). You will notice that this changes the vector's length but not its direction. 3. Multiply \(c \mathbf{v}\) by scalar \(a\), giving \(a(c \mathbf{v})\) with components \((ac v_x, ac v_y)\). 4. Multiply \(\mathbf{v}\) by scalar product \(a c\), giving \((a c) \mathbf{v}\) with components \((a c v_x, a c v_y)\). Both final vectors \(a(c \mathbf{v})\) and \((a c) \mathbf{v}\) have the same components, meaning they have the same length and direction. This confirms the property geometrically.