Problem 65 - A Graphical Approach to Precalculus with Limits

Question

Find the dot product of each pair of vectors. $$4 \mathbf{i}, 5 \mathbf{i}-9 \mathbf{j}$$

Step-by-Step Solution

Answer

The dot product is 20.

Step 1: Understanding Vectors

The first step in finding the dot product of two vectors is to understand the form of each vector. The first vector, \( \mathbf{a} = 4 \mathbf{i} \), has components \( (4, 0) \), since it only has an \( \mathbf{i} \) component and no \( \mathbf{j} \) component. The second vector, \( \mathbf{b} = 5 \mathbf{i} - 9 \mathbf{j} \), has components \( (5, -9) \).

Step 2: Dot Product Formula

The dot product \( \mathbf{a} \cdot \mathbf{b} \) of two vectors \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \) is calculated using the formula: \( \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \).

Step 3: Substitute Values

Substitute the components of the vectors \( \mathbf{a} = (4, 0) \) and \( \mathbf{b} = (5, -9) \) into the dot product formula: \( \mathbf{a} \cdot \mathbf{b} = 4 \times 5 + 0 \times (-9) \).

Step 4: Calculate Dot Product

Calculate the resulting expression: \( 4 \times 5 = 20 \) and \( 0 \times (-9) = 0 \). So the dot product is \( 20 + 0 = 20 \).

Key Concepts

Vectors

Vectors are mathematical objects used to represent quantities that have both magnitude and direction. They are quite useful in various fields such as physics, engineering, and computer graphics. In our exercise, we work with two-dimensional vectors expressed using unit vectors, often written as \( \mathbf{i} \) and \( \mathbf{j} \).
The vector \( 4 \mathbf{i} \) represents a direction aligned with the x-axis, having a length (or magnitude) of 4. Since there is no \( \mathbf{j} \) component, it does not extend in the y-direction at all. Meanwhile, \( 5 \mathbf{i} - 9 \mathbf{j} \) represents a vector moving 5 units right along the x-axis and 9 units down along the y-axis. Understanding these components is crucial when performing calculations such as the dot product.

Vector Components

The components of a vector refer to its projections along the coordinate axes. For a vector in the plane, these are typically along \( \mathbf{i} \) (the x-axis) and \( \mathbf{j} \) (the y-axis).
To find the vector components, one merely looks at the coefficients of \( \mathbf{i} \) and \( \mathbf{j} \). For \( 4 \mathbf{i} \), its components are \((4, 0)\). Here, 4 is the x-component, and 0 is the y-component, meaning it moves only horizontally.
For the vector \( 5 \mathbf{i} - 9 \mathbf{j} \), the components are \((5, -9)\). This vector has an x-component of 5 (indicating rightward movement) and a y-component of -9 (indicating a downward movement). Knowing each vector's components is vital in subsequent calculations.

Formula Application

In mathematics, formulas provide a standardized method to solve specific problems. The dot product formula we use here is a key tool for working with vectors: \( \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \). This formula combines the products of each pair of corresponding components from two vectors and sums them.
Using this formula allows us to determine how much two vectors "agree" in direction.
In our problem, we substitute the components of vectors \( \mathbf{a} = (4, 0) \) and \( \mathbf{b} = (5, -9) \) into the formula: \( 4 \times 5 + 0 \times (-9) \). This simplifies to \( 20 + 0 \), which results in the dot product of 20.

Problem Solving Steps

Solving vector problems involves a clear step-by-step approach. Let's break it down:

• Step 1: Identify the components of each vector. Here, \( \mathbf{a} \) has components (4,0), and \( \mathbf{b} \) has components (5,-9).
• Step 2: Apply the dot product formula \( \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \).
• Step 3: Substitute component values into the formula, resulting in \( 4 \times 5 + 0 \times (-9) \).
• Step 4: Perform the calculations to find the answer. The product simplifies to 20.

Following these steps ensures accuracy and clarity when finding the dot product between vectors.