Problem 68 - Precalculus

Question

Find the dot product: The dot product of vectors with \(n\) components is \(\left\langle a_{1}, a_{2}, \ldots, a_{n}\right\rangle \cdot\left\langle b_{1}, b_{2}, \ldots, b_{n}\right\rangle=a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n}\). $$\langle 1,0,-2,3\rangle \cdot\langle 5,2,3,1\rangle$$

Step-by-Step Solution

Answer

The dot product is 2.

Step 1: Identify Components

The components of the first vector are \( a_1 = 1 \), \( a_2 = 0 \), \( a_3 = -2 \), and \( a_4 = 3 \). The components of the second vector are \( b_1 = 5 \), \( b_2 = 2 \), \( b_3 = 3 \), and \( b_4 = 1 \).

Step 2: Multiply Corresponding Components

Calculate the products of corresponding components: 1. \( a_1 \cdot b_1 = 1 \cdot 5 = 5 \) 2. \( a_2 \cdot b_2 = 0 \cdot 2 = 0 \)3. \( a_3 \cdot b_3 = -2 \cdot 3 = -6 \)4. \( a_4 \cdot b_4 = 3 \cdot 1 = 3 \)

Step 3: Add the Products

Add the products calculated in the previous step: \( 5 + 0 + (-6) + 3 \) to get the dot product.

Step 4: Calculate the Sum

Perform the addition: \( 5 + 0 - 6 + 3 = 2 \).

Key Concepts

Vector Components

Vectors are entities that have both magnitude and direction. In a mathematical context, they are represented as an ordered list of numbers, known as components. For instance, consider the vectors used in the dot product calculation: \( \langle 1, 0, -2, 3 \rangle \) and \( \langle 5, 2, 3, 1 \rangle \). These numbers are the individual components.
Each component represents a projection of the vector along one dimension of space. For example:

• The first component "1" in \( \langle 1, 0, -2, 3 \rangle \) indicates the extent of the vector in the first direction.
• The "0" signifies no movement in the second direction.
• Similarly, "-2" and "3" show the vector's projection in the third and fourth dimensions.

Understanding vector components is essential, as they allow us to break down complex movements into simpler parts. This decomposition is the backbone of many applications in physics and engineering where vectors are used.

Vector Multiplication

Vector multiplication often refers to two main operations: the dot product and the cross product. In this case, we focus on the dot product. The dot product of two vectors is a scalar (a single number) obtained by multiplying corresponding components and adding them up.
For vectors like \( \langle 1, 0, -2, 3 \rangle \) and \( \langle 5, 2, 3, 1 \rangle \), the dot product is calculated as follows:

• Multiply the first components: \( 1 \times 5 = 5 \)
• Multiply the second components: \( 0 \times 2 = 0 \)
• Multiply the third components: \( -2 \times 3 = -6 \)
• Multiply the fourth components: \( 3 \times 1 = 3 \)

Then add up these products: \( 5 + 0 - 6 + 3 = 2 \). The result, \( 2 \), is the dot product. This operation is significant in physics, particularly in calculating the work done by a force. Additionally, the dot product is useful in determining the angle between vectors.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear equations. It is the framework within which operations like dot products are conducted. The study of linear algebra provides tools to solve systems of equations, transform geometric objects, and understand multi-dimensional spaces.
The dot product operation is a fundamental element of linear algebra. It helps us discover relationships between vectors in terms of their alignment and is crucial in defining concepts like orthogonal (perpendicular) vectors. Two vectors are orthogonal if their dot product equals zero.
Linear transformations and matrices, also key components of linear algebra, rely on vector operations. These transformations can scale, rotate, or otherwise manipulate vector quantities to suit various applications, from computer graphics to engineering and beyond.
By understanding linear algebra, learners unlock the ability to analyze and interpret a broad array of mathematical models. Whether designing algorithms, predicting data trends, or optimizing systems, linear algebra is a potent tool in a mathematician's toolkit.