Problem 11 - APEX Calculus

Question

Create your own vectors \(\vec{u}, \vec{v}\) and \(\vec{w}\) in \(\mathbb{R}^{2}\) and show that \(\vec{u} \cdot(\vec{v}+\vec{w})=\vec{u} \cdot \vec{v}+\vec{u} \cdot \vec{w}\)

Step-by-Step Solution

Answer

The equation holds true, demonstrating the distributive property of the dot product.

Step 1: Define the Vectors

Create your own vectors: let \( \vec{u} = \begin{pmatrix} 1 \ 2 \end{pmatrix}, \vec{v} = \begin{pmatrix} 3 \ 4 \end{pmatrix}, \) and \( \vec{w} = \begin{pmatrix} 5 \ 6 \end{pmatrix} \) in \( \mathbb{R}^{2} \). Write these vectors down for reference.

Step 2: Find Sum of Vectors \(\vec{v}\) and \(\vec{w}\)

Calculate the sum \( \vec{v} + \vec{w} \): \( \begin{pmatrix} 3 \ 4 \end{pmatrix} + \begin{pmatrix} 5 \ 6 \end{pmatrix} = \begin{pmatrix} 8 \ 10 \end{pmatrix} \). This represents the combined vector of \(\vec{v}\) and \(\vec{w}\).

Step 3: Compute Dot Product on Left-Hand Side

Find the dot product \(\vec{u} \cdot (\vec{v} + \vec{w})\): \( \begin{pmatrix} 1 \ 2 \end{pmatrix} \cdot \begin{pmatrix} 8 \ 10 \end{pmatrix} = 1 \cdot 8 + 2 \cdot 10 = 8 + 20 = 28 \).

Step 4: Compute Dot Products on Right-Hand Side

Calculate \(\vec{u} \cdot \vec{v}\) and \(\vec{u} \cdot \vec{w}\) separately:1. \(\vec{u} \cdot \vec{v} = \begin{pmatrix} 1 \ 2 \end{pmatrix} \cdot \begin{pmatrix} 3 \ 4 \end{pmatrix} = 1 \cdot 3 + 2 \cdot 4 = 3 + 8 = 11 \).2. \(\vec{u} \cdot \vec{w} = \begin{pmatrix} 1 \ 2 \end{pmatrix} \cdot \begin{pmatrix} 5 \ 6 \end{pmatrix} = 1 \cdot 5 + 2 \cdot 6 = 5 + 12 = 17 \).Add these results: \(11 + 17 = 28\).

Step 5: Compare Results

Observe that both sides of the equation equal 28, hence showing \( \vec{u} \cdot (\vec{v} + \vec{w}) = \vec{u} \cdot \vec{v} + \vec{u} \cdot \vec{w} \). This confirms the distributive property of the dot product.

Key Concepts

Distributive Property of the Dot Product

In the world of vectors, the distributive property is a fundamental concept. It states that if you have a vector multiplied by the sum of two other vectors, you can break it down into individual dot products. It's like distributing multiplication over addition. In mathematical terms, this property is expressed as:

• \( \vec{u} \cdot (\vec{v} + \vec{w}) = \vec{u} \cdot \vec{v} + \vec{u} \cdot \vec{w} \).

This essentially means that when you calculate the dot product of a vector \( \vec{u} \) with the sum of vectors \( \vec{v} \) and \( \vec{w} \), it's the same as adding the dot products \( \vec{u} \cdot \vec{v} \) and \( \vec{u} \cdot \vec{w} \).
In the exercise provided, the distributive property is verified by showing that the dot product of \( \vec{u} \) with the sum of \( \vec{v} \) and \( \vec{w} \) equals the sum of the dot products \( \vec{u} \cdot \vec{v} \) and \( \vec{u} \cdot \vec{w} \). The solution neatly confirms that this property holds true, making it a reliable tool when working with vector expressions.

Understanding Vector Addition

Vector addition is a straightforward yet powerful operation. It involves adding corresponding components of two vectors to form a new vector. Imagine you have two vectors \( \vec{v} = \begin{pmatrix} 3 \ 4 \end{pmatrix} \) and \( \vec{w} = \begin{pmatrix} 5 \ 6 \end{pmatrix} \). When you add them together, you simply add the first components, and then the second components like so:

• \( \vec{v} + \vec{w} = \begin{pmatrix} 3 \ 4 \end{pmatrix} + \begin{pmatrix} 5 \ 6 \end{pmatrix} = \begin{pmatrix} 8 \ 10 \end{pmatrix} \)

This process combines the two vectors into one, summed vector.
Vector addition is commutative, meaning the order in which you add vectors doesn't matter. So \( \vec{v} + \vec{w} \) will give you the same result as \( \vec{w} + \vec{v} \). This property is essential when handling multiple vector operations, as it allows flexibility in how calculations are structured.
In our exercise, you see vector addition applied when \( \vec{v} \) and \( \vec{w} \) are combined into a single vector to verify the distributive property.

Vector Multiplication and the Dot Product

Vector multiplication can take different forms, but when we talk about the dot product, it's a specific kind of multiplication. The dot product is a scalar that results from multiplying corresponding components of two vectors and then summing those products.
For example, with vectors \( \vec{u} = \begin{pmatrix} 1 \ 2 \end{pmatrix} \) and \( \vec{v} = \begin{pmatrix} 3 \ 4 \end{pmatrix} \), the dot product calculation is as follows:

• \( \vec{u} \cdot \vec{v} = 1 \cdot\ 3 + 2 \cdot\ 4 = 3 + 8 = 11 \)

This operation gives a single numerical result, representing how closely the two vectors align with each other.
The dot product is commutative, much like vector addition. Therefore, \( \vec{u} \cdot \vec{v} \) will yield the same value as \( \vec{v} \cdot \vec{u} \).
In the exercise, the dot product is crucial for confirming the distributive property. First, the dot product involving the sum of vectors is calculated, and then individual dot products are used to confirm the equation.