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Chapter 8: Problem 71

Find the dot product for each pair of vectors. $$\langle 6,-1\rangle,\langle 2,5\rangle$$

Short Answer

Expert verified
The dot product is 7.

Step by step solution

01

- Understand the Dot Product Formula

The dot product of two vectors \(\textbf{a} = \langle a_1, a_2 \rangle\) and \(\textbf{b} = \langle b_1, b_2 \rangle\) is given by the formula: \[ \textbf{a} \bullet \textbf{b} = a_1 \times b_1 + a_2 \times b_2 \]
02

- Identify the Components of the Vectors

Identify the components of the given vectors. For vector \(\textbf{a} = \langle 6, -1 \rangle\), the components are: \( a_1 = 6 \) and \( a_2 = -1 \). For vector \(\textbf{b} = \langle 2, 5 \rangle\), the components are: \( b_1 = 2 \) and \( b_2 = 5 \).
03

- Multiply Corresponding Components

Multiply the corresponding components of the vectors: \( a_1 \times b_1 = 6 \times 2 = 12 \) and \( a_2 \times b_2 = -1 \times 5 = -5 \).
04

- Add the Products

Add the results from Step 3 to get the dot product: \(12 + (-5) = 7\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Operation
Vectors are essential in both physics and mathematics. They provide a way to describe quantities that have both direction and magnitude. When working with vectors, there are several operations you can perform. One of the fundamental operations in vector analysis is the dot product.

The dot product (also known as the scalar product) combines two vectors into a single scalar value. This value can help determine projections between the vectors and their relative angles.
Component Multiplication
To find the dot product, we use a process called component multiplication. Each vector has components that you multiply together.

For example, consider the vectors \(\textbf{a} = \langle 6,-1\rangle\) and \(\textbf{b} = \langle 2,5\rangle\). We start by identifying their components: \(a_1 = 6, a_2 = -1\) and \(b_1 = 2, b_2 = 5\).

We multiply the corresponding components of the vectors: \(a_1 \times b_1\) and \(a_2 \times b_2\). Performing these operations gives us \(6 \times 2 = 12\) and \(-1 \times 5 = -5\). Calculating these products separately helps to simplify the final step.
Vector Algebra
Vector algebra involves performing operations like addition, subtraction, and dot product on vectors. These operations provide insights into the relationships between vectors.

In our exercise, we found the dot product using the following steps:
  • Identified the components of vectors \(\textbf{a}\) and \(\textbf{b}\).
  • Multiplied corresponding components: \(a_1 \times b_1\) and \(a_2 \times b_2\).
  • Added the resulting products.
Combining these steps gives us the dot product: \(12 + (-5) = 7\).

Understanding these basic operations is crucial for studying more complex concepts in vector algebra, such as projections and cross products.

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