Problem 35 - Algebra and Trigonometry

Question

33-36 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 dot product is distributive over vector addition: \((\mathbf{u}+\mathbf{v}) \cdot \mathbf{w}=\mathbf{u} \cdot \mathbf{w}+\mathbf{v} \cdot \mathbf{w}\).

Step 1: Understand the Problem

We need to prove the distributive property of the dot product over vector addition, which states \((\mathbf{u} + \mathbf{v}) \cdot \mathbf{w} = \mathbf{u} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{w}\). This means we have to show that the dot product distributes over addition of vectors.

Step 2: Recall the Definition of Dot Product

The dot product of two vectors \(\mathbf{a} = (a_1, a_2, a_3)\) and \(\mathbf{b} = (b_1, b_2, b_3)\) is defined as \(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\). We'll apply this definition to prove the property.

Step 3: Apply the Dot Product Definition to the Left Side

Let \(\mathbf{u} = (u_1, u_2, u_3)\), \(\mathbf{v} = (v_1, v_2, v_3)\), and \(\mathbf{w} = (w_1, w_2, w_3)\). Compute \((\mathbf{u} + \mathbf{v}) \cdot \mathbf{w}\): \((\mathbf{u} + \mathbf{v}) \cdot \mathbf{w} = ((u_1 + v_1), (u_2 + v_2), (u_3 + v_3)) \cdot (w_1, w_2, w_3)\). So it becomes: \((u_1 + v_1)w_1 + (u_2 + v_2)w_2 + (u_3 + v_3)w_3\).

Step 4: Expand the Expression

Use distribution for each multiplication: \((u_1 + v_1)w_1 = u_1w_1 + v_1w_1\), \((u_2 + v_2)w_2 = u_2w_2 + v_2w_2\), \((u_3 + v_3)w_3 = u_3w_3 + v_3w_3\). Thus, the full expression expands to \(u_1w_1 + v_1w_1 + u_2w_2 + v_2w_2 + u_3w_3 + v_3w_3\).

Step 5: Group the Like Terms

Group terms involving \(\mathbf{u}\) and \(\mathbf{v}\): \((u_1w_1 + u_2w_2 + u_3w_3) + (v_1w_1 + v_2w_2 + v_3w_3)\). This matches the form \(\mathbf{u} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{w}\).

Step 6: Conclude the Proof

We have shown \((\mathbf{u} + \mathbf{v}) \cdot \mathbf{w} \) is equal to \( \mathbf{u} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{w} \). Therefore, the dot product is distributive over vector addition, as required.

Key Concepts

Dot Product

The dot product is a fundamental concept in vector operations. It provides a way to multiply two vectors and results in a scalar.
To compute the dot product of two vectors, such as \(\mathbf{a} = (a_1, a_2, a_3)\) and \(\mathbf{b} = (b_1, b_2, b_3)\), we multiply corresponding components and sum them up:

• \(a_1 \cdot b_1\)
• \(a_2 \cdot b_2\)
• \(a_3 \cdot b_3\)

This results in \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \).
The dot product is also known as the scalar product because it results in a single scalar value, rather than a vector. This product measures how much one vector goes in the direction of another. If two vectors are perpendicular, their dot product is zero, which shows they have no component of one in the direction of the other.
Dot products have many applications in physics, computer graphics, and other fields that involve vector mathematics.

Distributive Property

The distributive property is a crucial rule in mathematics which applies to both numbers and vector operations. It tells how to distribute a common factor, like in expansion tasks. In the context of vectors and the dot product, this property states:

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

This means that the dot product distributes over vector addition.
To understand this better, think about how you might expand an expression such as \((a + b) \times c\), ending up with \(aa + bc\). Similarly, when dealing with vectors, you perform the dot product operation by distributing \(\mathbf{w}\) through the parentheses next to \(\mathbf{u}\) and \(\mathbf{v}\).
This results in adding the individual dot products \(\mathbf{u} \cdot \mathbf{w}\) and \(\mathbf{v} \cdot \mathbf{w}\).
This expansion and grouping process shows that the dot product is indeed distributive over vector addition, which is fundamental in simplifying many vector calculations.

Vector Addition

Vector addition is a basic operation involving combining two or more vectors to produce a sum vector. This operation is performed component-wise, meaning each component of the vectors is added together. For instance, consider two vectors \( \mathbf{u} = (u_1, u_2, u_3) \) and \( \mathbf{v} = (v_1, v_2, v_3) \). Their sum is computed as:

• \( u_1 + v_1 \)
• \( u_2 + v_2 \)
• \( u_3 + v_3 \)

Giving the result \( \mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2, u_3 + v_3) \).
The result of vector addition is also a vector, and it geometrically represents the diagonal of the parallelogram formed by \(\mathbf{u}\) and \(\mathbf{v}\) when placed tail to tail.
This property is utilized in the distributive property of the dot product, where we first add vectors \(\mathbf{u}\) and \(\mathbf{v}\), then compute the dot product with another vector \(\mathbf{w}\). Understanding vector addition helps in visualizing how vectors combine and interact in physical space, making it a vital concept in vector mathematics.