Problem 41 - Algebra and Trigonometry

Question

Let \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be vectors, and let \(a\) be a scalar. Prove the given property. $$ (\mathbf{u}+\mathbf{v}) \cdot \mathbf{w}=\mathbf{u} \cdot \mathbf{w}+\mathbf{v} \cdot \mathbf{w} $$

Step-by-Step Solution

Answer

The property is proven by distributing the dot product onto the sum and confirming each component matches individually.

Step 1: Understand the Distributive Property

The problem asks us to prove the distributive property of the dot product over vector addition. This property states that for any vectors \(\mathbf{u}, \mathbf{v}, \mathbf{w}\), the dot product of a sum with another vector equals the sum of the individual dot products. The expression \((\mathbf{u}+\mathbf{v}) \cdot \mathbf{w}\) should result in the same value as \(\mathbf{u} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{w}\).

Step 2: Break Down the Vectors into Components

Assume the vectors \(\mathbf{u}, \mathbf{v}, \mathbf{w}\) have components in \(\mathbb{R}^n\), i.e., \(\mathbf{u} = (u_1, u_2, \ldots, u_n)\), \(\mathbf{v} = (v_1, v_2, \ldots, v_n)\), and \(\mathbf{w} = (w_1, w_2, \ldots, w_n)\). The dot product of two vectors \(\mathbf{a} = (a_1, a_2, \ldots, a_n)\) and \(\mathbf{b} = (b_1, b_2, \ldots, b_n)\) is defined as \(\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + \ldots + a_n b_n\).

Step 3: Express \((\mathbf{u} + \mathbf{v}) \cdot \mathbf{w}\)

Using the component-wise addition of vectors, express the vector \((\mathbf{u} + \mathbf{v}) = (u_1 + v_1, u_2 + v_2, \ldots, u_n + v_n)\). Then compute the dot product with \(\mathbf{w}\): \[(\mathbf{u} + \mathbf{v}) \cdot \mathbf{w} = (u_1 + v_1)w_1 + (u_2 + v_2)w_2 + \ldots + (u_n + v_n)w_n.\]

Step 4: Distribute Each Term

Distribute \(w_i\) across each term's addition, applying the distributive law of arithmetic, which gives: \[(u_1 + v_1)w_1 = u_1 w_1 + v_1 w_1, (u_2 + v_2)w_2 = u_2 w_2 + v_2 w_2, \ldots, (u_n + v_n)w_n = u_n w_n + v_n w_n.\]

Step 5: Combine the Terms

After distributing, express the entire dot product as a sum of two separate sums: \[(u_1 w_1 + v_1 w_1) + (u_2 w_2 + v_2 w_2) + \ldots + (u_n w_n + v_n w_n)\]which simplifies to: \[(u_1 w_1 + u_2 w_2 + \ldots + u_n w_n) + (v_1 w_1 + v_2 w_2 + \ldots + v_n w_n).\]

Step 6: Express the Summands as Dot Products

Notice that each of these two sums is the dot product of one of the original vectors with \(\mathbf{w}\). Therefore, rewrite the expression as: \[\mathbf{u} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{w}.\]This shows that \[(\mathbf{u} + \mathbf{v}) \cdot \mathbf{w} = \mathbf{u} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{w},\]concluding the proof of the distributive property for the dot product.

Key Concepts

Vector Addition

Vector addition is the process of adding two or more vectors together to obtain another vector. Think of it as a way of combining the directions and magnitudes of vectors. If you imagine vectors as arrows, adding them involves placing them in a sequence, one after another, and the resultant vector is drawn from the start of the first vector to the end of the last one.

In mathematical terms, if we have two vectors \( \mathbf{u} \) and \( \mathbf{v} \) each with components in \( \mathbb{R}^n \), their addition is performed component-wise:

• \( \mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2, \ldots, u_n + v_n) \)

This means that for each component of the resulting vector, we add the corresponding components of \( \mathbf{u} \) and \( \mathbf{v} \). Vector addition is commutative and associative, meaning you can add vectors in any order and still get the same result.

Dot Product

The dot product, also known as the "scalar product," is a way of multiplying two vectors that result in a scalar value. It provides insight into the relationship between the directions and lengths of vectors. The dot product is especially useful in physics and engineering to calculate things like work done by a force.

The dot product of two vectors \( \mathbf{a} = (a_1, a_2, \ldots, a_n) \) and \( \mathbf{b} = (b_1, b_2, \ldots, b_n) \) is calculated as:

\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n \]

This formula results in a single number, which is the sum of the products of their respective components. The dot product is commutative, meaning \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \). If the dot product is zero, it indicates that the vectors are orthogonal, or at a right angle to each other.

Vector Components

Vector components are the projections of a vector along the axes of a coordinate system. In essence, they express the vector in terms of its influence or stretch in each coordinate direction. Imagine a vector broken down, step by step, into parts that lie strictly along each axis.

A vector \( \mathbf{u} \) in \( \mathbb{R}^n \) can be represented by its components as: \( \mathbf{u} = (u_1, u_2, \ldots, u_n) \).

• These components \( u_1, u_2, \ldots, u_n \) express how far and in which direction the vector \( \mathbf{u} \) extends along the axes.

By using vector components, we handle calculations more easily and can systematically apply mathematical operations like vector addition and the dot product. This makes vector arithmetic consistent and efficient.

Proof by Component Method

Proof by component method is a step-by-step mathematical technique used to validate properties of vectors. It involves breaking down vectors into their basic elements, or components, and then verifying the property at each component level.

For the distributive property of the dot product, this method involves several stages. Consider vectors \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \):

• First, write each vector in component form:
• \( \mathbf{u} = (u_1, u_2, \ldots, u_n) \), \( \mathbf{v} = (v_1, v_2, \ldots, v_n) \), \( \mathbf{w} = (w_1, w_2, \ldots, w_n) \).
• Next, substitute these component forms into the expression \((\mathbf{u} + \mathbf{v}) \cdot \mathbf{w} \).

Compute this dot product, and you'll see how each component follows the distributive law: \( (u_i + v_i)w_i = u_iw_i + v_iw_i \). This individually checked equality establishes the general property across the entire vector, verifying it for all components.

This logical breakdown not only proves properties succinctly but also builds a strong foundational understanding of vector operations.