Calculating Dot Product Of Vectors A Step By Step Guide

When dealing with vectors, understanding the dot product is crucial. In this article, we will delve into calculating the dot product of vectors, specifically addressing the problem: If ${\vec{a} = [1, 2, -1]}$ and ${\vec{b} = [-1, 2, 1]}$, find ${\vec{a} \cdot (\vec{a} + \vec{b})}$. This involves both vector addition and the dot product operation, which are fundamental concepts in linear algebra and vector calculus. Vectors are mathematical objects that have both magnitude and direction, making them essential in physics, engineering, and computer graphics. The dot product, also known as the scalar product, provides a way to multiply two vectors and obtain a scalar value, which can be used to determine the angle between the vectors, among other things. The dot product is a scalar value representing the projection of one vector onto another, scaled by the magnitude of the second vector. It's a fundamental operation in linear algebra, with applications ranging from physics to computer graphics. Understanding how to perform vector operations such as addition and dot product is essential for various scientific and engineering applications. Before we tackle the main problem, let's briefly recap the basic principles of vector addition and dot product calculation.

Vector Addition

To add two vectors, you simply add their corresponding components. For example, if we have two vectors ${\vec{u} = [u_1, u_2, u_3]}$ and ${\vec{v} = [v_1, v_2, v_3]}$, their sum ${\vec{u} + \vec{v}}$ is calculated as:

${ \vec{u} + \vec{v} = [u_1 + v_1, u_2 + v_2, u_3 + v_3] }$

This operation results in a new vector whose components are the sums of the corresponding components of the original vectors. Vector addition follows the parallelogram law, where the sum represents the diagonal of the parallelogram formed by the two vectors. Vector addition is a fundamental operation, allowing us to combine vectors to represent net forces, displacements, or velocities. The resultant vector captures the combined effect of the individual vectors. Vector addition is both commutative and associative, meaning the order in which vectors are added doesn't affect the result, and vectors can be grouped in any way during addition. This property is crucial in various applications, such as calculating the resultant force in physics or determining the overall displacement in engineering problems.

Dot Product

The dot product of two vectors ${\vec{u}}$ and ${\vec{v}}$ is a scalar value obtained by multiplying their corresponding components and summing the results. Using the same vectors ${\vec{u}}$ and ${\vec{v}}$ as before, the dot product ${\vec{u} \cdot \vec{v}}$ is calculated as:

${ \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3 }$

The dot product has several important properties. It is commutative, meaning ${\vec{u} \cdot \vec{v} = \vec{v} \cdot \vec{u}}$. It is also distributive over vector addition, meaning ${\vec{u} \cdot (\vec{v} + \vec{w}) = \vec{u} \cdot \vec{v} + \vec{u} \cdot \vec{w}}$. Furthermore, the dot product can be used to find the angle ${\theta}$ between two vectors using the formula:

${ \vec{u} \cdot \vec{v} = ||\vec{u}|| \cdot ||\vec{v}|| \cdot \cos(\theta) }$

Where ${||\vec{u}||}$ and ${||\vec{v}||}$ represent the magnitudes (or lengths) of the vectors ${\vec{u}}$ and ${\vec{v}}$, respectively. The dot product is a powerful tool for determining the relationship between vectors, including their orthogonality (when the dot product is zero) and the angle between them. Understanding the dot product is essential for various applications, such as calculating work done by a force, projecting one vector onto another, and determining the orientation of surfaces in computer graphics. The dot product also provides a way to measure the similarity between two vectors, with a higher dot product indicating a greater alignment. This property is widely used in machine learning and data analysis for tasks such as clustering and recommendation systems.

Given the vectors ${\vec{a} = [1, 2, -1]}$ and ${\vec{b} = [-1, 2, 1]}$, we want to find ${\vec{a} \cdot (\vec{a} + \vec{b})}$. This involves two steps:

1. Vector Addition: First, we need to find the sum of vectors ${\vec{a}}$ and ${\vec{b}}$.
2. Dot Product: Then, we calculate the dot product of ${\vec{a}}$ with the resulting vector from step 1.

Step 1: Vector Addition

To find ${\vec{a} + \vec{b}}$, we add the corresponding components of ${\vec{a}}$ and ${\vec{b}}$:

${ \vec{a} + \vec{b} = [1 + (-1), 2 + 2, -1 + 1] = [0, 4, 0] }$

So, the sum of the vectors ${\vec{a}}$ and ${\vec{b}}$ is the vector ${[0, 4, 0]}$. This resultant vector represents the combined effect of the individual vectors, capturing both magnitude and direction. Vector addition is a fundamental operation in physics and engineering, where it's used to calculate the resultant force, displacement, or velocity. The resultant vector ${[0, 4, 0]}$ indicates that the combined effect of ${\vec{a}}$ and ${\vec{b}}$ is primarily in the y-direction, with no component in the x or z directions. Understanding vector addition is crucial for solving problems involving multiple forces or motions, allowing us to predict the net effect on an object or system.

Step 2: Dot Product

Next, we need to find the dot product of ${\vec{a}}$ and the vector ${\vec{a} + \vec{b}}$, which we found to be ${[0, 4, 0]}$. The dot product ${\vec{a} \cdot (\vec{a} + \vec{b})}$ is calculated as:

${ \vec{a} \cdot (\vec{a} + \vec{b}) = [1, 2, -1] \cdot [0, 4, 0] = (1 \cdot 0) + (2 \cdot 4) + (-1 \cdot 0) = 0 + 8 + 0 = 8 }$

Therefore, ${\vec{a} \cdot (\vec{a} + \vec{b}) = 8}$. This scalar value represents the projection of ${\vec{a}}$ onto the vector ${\vec{a} + \vec{b}}$, scaled by the magnitude of ${\vec{a} + \vec{b}}$. The dot product is a fundamental tool for determining the relationship between vectors, including their orthogonality and the angle between them. In this case, the dot product of 8 indicates that ${\vec{a}}$ and ${\vec{a} + \vec{b}}$ have a significant alignment, as a non-zero dot product suggests that the vectors are not orthogonal. The dot product is widely used in various applications, such as calculating work done by a force, projecting one vector onto another, and determining the orientation of surfaces in computer graphics. Understanding the dot product is essential for solving problems involving vector projections, work calculations, and angular relationships between vectors.

After performing the vector addition and dot product, we find that ${\vec{a} \cdot (\vec{a} + \vec{b}) = 8}$. Therefore, the correct answer is:

D) 8

This exercise demonstrates the importance of understanding vector operations, particularly vector addition and the dot product. These concepts are fundamental in various fields, including physics, engineering, and computer science. Vector operations are essential tools for solving problems involving forces, velocities, and spatial relationships. The ability to manipulate vectors and calculate their relationships is crucial for many scientific and engineering applications. Understanding the geometric interpretations of vector operations, such as the dot product representing a projection and the cross product representing an area, provides a deeper insight into their significance. The dot product, in particular, is a powerful tool for determining the angle between vectors, checking for orthogonality, and calculating work done by a force. The problem we solved illustrates how these concepts can be applied to solve practical problems, emphasizing the importance of mastering these fundamental vector operations.