Problem 7 - Essential Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The dot product \( \mathbf{a} \cdot \mathbf{b} = 1 \).

Step 1: Recall the Dot Product Formula

The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \), where \( a_1, a_2, a_3 \) and \( b_1, b_2, b_3 \) are the components of \( \mathbf{a} \) and \( \mathbf{b} \) respectively.

Step 2: Identify the Vector Components

Write the vectors \( \mathbf{a} \) and \( \mathbf{b} \) as component vectors: \( \mathbf{a} = 2\mathbf{i} + \mathbf{j} = (2, 1, 0) \) and \( \mathbf{b} = \mathbf{i} - \mathbf{j} + \mathbf{k} = (1, -1, 1) \).

Step 3: Apply the Dot Product Formula

Substitute the components into the dot product formula: \( \mathbf{a} \cdot \mathbf{b} = 2 \cdot 1 + 1 \cdot (-1) + 0 \cdot 1 \).

Step 4: Perform the Calculations

Calculate each term: \( 2 \cdot 1 = 2 \), \( 1 \cdot (-1) = -1 \), and \( 0 \cdot 1 = 0 \). Add these results to find the dot product: \( 2 - 1 + 0 = 1 \).

Step 5: State the Result

Therefore, the dot product \( \mathbf{a} \cdot \mathbf{b} \) is 1.

Key Concepts

Vector Components

Vectors are mathematical objects that have both magnitude and direction. In order to perform calculations like the dot product, it's essential to understand a vector's components.
Vector components refer to the values of a vector along the coordinate axes. For example, in a 3D space, any vector can be broken down into components along the x, y, and z axes. For the vector \( \mathbf{a} = 2\mathbf{i} + \mathbf{j} \), this means it can be expressed as the ordered triple \((2, 1, 0)\). Similarly, \( \mathbf{b} = \mathbf{i} - \mathbf{j} + \mathbf{k} \) has components \((1, -1, 1)\). This breakdown helps us easily calculate the dot product and interpret the vector in spatial terms.
When expressing a vector in terms of its components:

• Each unit vector (i.e., \( \mathbf{i} \), \( \mathbf{j} \), \( \mathbf{k} \)) corresponds to one axis (x, y, z).
• The coefficients of these unit vectors are the components of the vector.
• Organizing these components in a tuple form clarifies the representation.

Knowing how to identify and use vector components is crucial when working with vector operations.

Dot Product Formula

The dot product is a specific operation you can perform on two vectors, culminating in a scalar (a single number) instead of another vector. This operation is useful in various applications, such as finding the angle between vectors or determining if two vectors are orthogonal.
To compute the dot product, use the formula:
\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \] where \( a_1, a_2, a_3 \) and \( b_1, b_2, b_3 \) are components of the vectors \( \mathbf{a} \) and \( \mathbf{b} \), respectively.
When carrying out the dot product:

• Multiply corresponding components from each vector.
• Add up all these individual products.
• The resulting sum is the dot product, a scalar value representing the vectors' directional similarity.

In our example, by substituting the components, the dot product \( \mathbf{a} \cdot \mathbf{b} \) was calculated by \( 2 \cdot 1 + 1 \cdot (-1) + 0 \cdot 1 \), ultimately resulting in 1. By deeply understanding this formula, you can tackle a broad spectrum of vector-based problems.

Vector Operations

Vector operations include a variety of calculations you can perform on vectors, each requiring a solid understanding of vector components. Common vector operations include addition, subtraction, scalar multiplication, and the dot product.
These operations allow us to manipulate vectors in meaningful ways, solving physics problems, performing graphic transformations, and more.
Here's a quick breakdown of some key vector operations:

• **Addition and Subtraction:** Combine or differentiate vectors by adding or subtracting their components. For example, \( \mathbf{a} + \mathbf{b} \) involves adding corresponding components from both vectors.
• **Scalar Multiplication:** Multiply each component of a vector by a scalar (a single number), altering the vector's magnitude but not its direction.
• **Dot Product:** Explored extensively above, this gives a scalar that tells us how much two vectors align with each other.

Familiarity with these operations enriches problem-solving capabilities when dealing with vectors in any dimension. Understanding vector operations is crucial for performing and interpreting calculations, as seen in physics and engineering applications.