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

Short answer

Based on the step by step solution above, answer the following question: Question: Prove the vector property \((a+c) \mathbf{v}=a \mathbf{v}+c \mathbf{v}\) using components. Answer: The vector property is proved by expressing the vector \(\textbf{v}\) in terms of its components, performing scalar-vector multiplication for both sides of the equation, and comparing the resulting components. After the distributions and simplifications, we can observe that the components in the \(i\) and \(j\) directions are equal, and we can conclude that \((a+c) \mathbf{v}=a \mathbf{v}+c \mathbf{v}\). Additionally, a geometrical sketch confirms the distributive property of scalar-vector multiplication.

Step-by-step solution

Write vector v in terms of its components

We know \(\mathbf{v}\) is a vector in the \(xy\) plane. Let's denote its components as \(v_x\) and \(v_y\). So, we can write the vector \(\mathbf{v}\) as: $$\mathbf{v} = v_x\mathbf{i} + v_y\mathbf{j}$$

Perform scalar multiplication for both sides of the equation

Now, we perform scalar-vector multiplication for both sides of the equation \((a+c) \mathbf{v}=a \mathbf{v}+c \mathbf{v}\) based on the components of \(\mathbf{v}\). Left side of the equation: $$(a+c)\mathbf{v} = (a+c)(v_x\mathbf{i}+v_y\mathbf{j})$$ Right side of the equation: $$a\mathbf{v}+c\mathbf{v} = a(v_x\mathbf{i}+v_y\mathbf{j}) + c(v_x\mathbf{i}+v_y\mathbf{j})$$

Distribute the scalars and simplify

Distribute the scalars for both sides of the equation and simplify. Left side of the equation: $$(a+c)\mathbf{v} = ((a+c)v_x)\mathbf{i} + ((a+c)v_y)\mathbf{j}$$ Right side of the equation: $$a\mathbf{v}+c\mathbf{v} = (av_x+cv_x)\mathbf{i} + (av_y+cv_y)\mathbf{j}$$

Comparison and proof

Comparing both sides of the equation, we can observe that the components in the \(i\) and \(j\) directions are equal. For the \(i\) direction: $$(a+c)v_x = av_x + cv_x$$ For the \(j\) direction: $$(a+c)v_y = av_y + cv_y$$ Since the components are equal, we can conclude that: $$(a+c) \mathbf{v}=a \mathbf{v}+c \mathbf{v}$$

Geometrical sketch

To illustrate the property geometrically, let's consider a vector \(\mathbf{v}\). Let's represent the scalar-vector products \(a\mathbf{v}\) and \(c\mathbf{v}\) as two vectors originating from the same point, and create a parallelogram using these vectors. Now let's represent the vector \((a+c)\mathbf{v}\) from the same starting point. We can observe that the vector sum \(a\mathbf{v} + c\mathbf{v}\) (diagonal of parallelogram) is equal to the vector \((a+c)\mathbf{v}\). This geometric sketch visually demonstrates the proved property.