Problem 8 - Essential Calculus Early Transcendentals

Question

2-10 Find \(\mathbf{a} \cdot \mathbf{b}\) $$\mathbf{a}=3 \mathbf{i}+2 \mathbf{j}-\mathbf{k}, \quad \mathbf{b}=4 \mathbf{i}+5 \mathbf{k}$$

Step-by-Step Solution

Answer

The dot product \( \mathbf{a} \cdot \mathbf{b} = 7 \).

Step 1: Understand the Dot Product Formula

The dot product of two vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k} \) is calculated as \( \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 \).

Step 2: Identify Components

Identify the components from both vectors:- \( \mathbf{a} = 3 \mathbf{i} + 2 \mathbf{j} - \mathbf{k} \) implies \( a_1 = 3, a_2 = 2, a_3 = -1 \)- \( \mathbf{b} = 4 \mathbf{i} + 5 \mathbf{k} \) implies \( b_1 = 4, b_2 = 0, b_3 = 5 \)

Step 3: Calculate Each Product

Multiply corresponding components of \( \mathbf{a} \) and \( \mathbf{b} \):- \( a_1 b_1 = 3 \times 4 = 12 \)- \( a_2 b_2 = 2 \times 0 = 0 \)- \( a_3 b_3 = (-1) \times 5 = -5 \)

Step 4: Sum the Results

Sum the results from step 3 to find the dot product:\[ \mathbf{a} \cdot \mathbf{b} = 12 + 0 - 5 = 7 \]

Key Concepts

Vectors

Vectors are mathematical objects that have both magnitude and direction. Think of them like arrows pointing in a specific direction and with a certain length. This makes them very useful in representing physical quantities such as velocity, force, and displacement.

• Notation: Vectors are usually noted with bold letters, like \( \mathbf{a} \) or \( \mathbf{b} \).
• Direction and Magnitude: The direction shows where the vector is pointing, and the magnitude indicates how long the vector is. Mathematically, the magnitude or length of a vector can be found using the Pythagorean theorem when the components of the vector are known.
• Components: Vectors can exist in two-dimensional or three-dimensional spaces. In three-dimensional space, they are commonly expressed in terms of the unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) that point along the x, y, and z axes respectively.

Understanding vectors is the first step in dealing with more complex operations like finding the dot product, which involves these components.

Vector Components

Vector components are the building blocks of vectors. They are crucial for performing calculations since they allow you to break down vectors into simpler parts that align with each of the coordinate axes.
How to Determine Vector Components:

• Each vector component represents how far the vector extends along a particular axis.
• For instance, if a vector is expressed as \( 3\mathbf{i} + 2\mathbf{j} - \mathbf{k} \), the components are \(3\) along the x-axis, \(2\) along the y-axis, and \(-1\) along the z-axis.

By using components, vectors can be easily added, subtracted, and importantly, used in operations such as the dot product. These components help in performing calculations by acting similarly to simple numbers, but in a directional context.

Scalar Product

The scalar product, or dot product, is a way to multiply two vectors to get a scalar. This means that while vectors are used in calculation, the final answer is just a single number, called a scalar.
Understanding the Dot Product Formula:

• The formula for the dot product of two vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k} \) is \( \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 \).
• It is important to carefully pair and multiply corresponding components from both vectors, as illustrated in the exercise's solution.
• The result is a measure of how much one vector "projects" onto the other, capturing the idea of their directional alignment.

In our particular exercise, breaking down the dot product into its components and summing them accurately provided the correct scalar product, which highlights the importance of understanding components and their interactions.