Chapter 2: Introduction to Vectors

Table of Contents

2.1 Introduction

Vectors are a fundamental concept in physics which describe quantities that have both magnitude and direction. This definition cannot be stressed enough. Unlike scalars, which only have magnitude (such as temperature, mass, or time), vectors allow us to describe physical quantities that have both a size and an associated direction, such as velocity, acceleration, and force.

We often denote vectors by an arrow on top of a letter. For example, the velocity vector is usually denoted $\vec{v}.$ This is how we will signify vectors for the remainder of this course (as well as in TruPHY 201 and 301). However, it is useful to know that some textbooks and online resources will denote vectors using boldface. Historically, this was useful because of limitations in technology, but, at this point, it is much more clear to just use the vector arrow.

Furthermore, it is useful to know that sometimes we only wish to know the magnitude of a vector. When this is the case, we can simply add in absolute value bars as these remove information regarding directionality. When we do so, it looks like this: $|\vec{v}|.$

2.2 Vectors Help Us Describe and Understand Motion

So why do we need vectors? In physics, vectors provide a powerful tool for describing and understanding motion and forces in the world around us. For example, if we know the velocity of an object (its speed and direction), we can use vectors to calculate its position at any given time. Similarly, if we know the forces acting on an object (their magnitudes and directions), we can use vectors to calculate the object’s motion.

Vectors are best represented as arrows which clearly depict both magnitude and direction.

Muddy point: Many beginner physics students make the mistake of thinking that speed is a vector. After all, when you are driving down the road, you often will associate the direction you are traveling with your speed. Thus, with a magnitude and direction, speed must be a vector! Not so fast. Speed is a scalar only. It is how fast you are going, nothing more. The moment you attach a direction to a speed, it becomes a velocity. Take the time to get acquainted with this terminology as it will benefit you a great deal down the road.

2.3 Applications of Vectors

Vectors are used in a wide range of everyday applications, from navigation and weather forecasting to sports and engineering. For example:

In navigation, vectors are used to describe the direction and speed of wind and currents, which can help sailors find their way at sea.
In weather forecasting, vectors are used to describe the movement of air masses, which can help meteorologists predict the direction and strength of storms.
Sports analysis uses vectors to describe the velocity and acceleration of balls and athletes, which can help coaches review performance and improve technique.
In engineering, vectors are used to describe the forces acting on structures, such as bridges and buildings, which can help engineers design safer and more durable structures.

2.4 Vector Addition and Subtraction

When adding vectors, you can’t just perform normal addition or subtraction on the magnitudes. You need to account for the direction as well. We will discuss a more useful method for doing so in the next chapter. However, it is first essential to gain a qualitative understanding of vector addition (and subtraction) through the tip-to-tail method.

The method works like this:

2.4.1 Addition

When adding two vectors, $\vec{A} \text{ and } \vec{B}$ , Place the “tail” of $\vec{B}$ at the “tip” of $\vec{A}$ without rotating either one. Once this is complete, draw a new arrow connecting the “tail” of $\vec{A}$ to the “tip” of $\vec{B}$ . If you have done this correctly, you will notice that the new arrow forms a triangle with the other two vectors. We call this new arrow the resultant vector. This is important terminology, so be sure to memorize it. We can mathematically represent vector addition as:

$\vec{A}+\vec{B}$

Vector addition.

In the figure above, you can clearly visualize the tip-to-tail method in action. Note the way that the resultant vector is drawn and compare that with the image below when the exacct same vectors are subtracted one from the other.

2.4.2 Subtraction

On the other hand, when subtracting two vectors, all you are really doing is flipping the direction of the subtracted vector. From there, you carry on with the tip-to-tail method as usual. In other words, take the example of $\vec{A}-\vec{B}.$ Keep $\vec{A}$ as it is. Bring over vector $\vec{B}$ but flip it’s direction so it is pointing opposite to the way it was originally. Then place the “tail” of $\vec{B}$ to the “tip” of $\vec{A}.$ Draw the resultant vector and you’ve done it. That’s $\vec{A}-\vec{B}.$ Another common mathematical representation of vector subtraction is to just make it vector addition with a negative sign:

$\vec{A}+(-\vec{B})$

Vector subtraction.

This figure shows clearly how the resultant vector changes when you subtract $\vec{B}$ rather than add it. Note how the negative sign is equivalent to a change in direction, the same as rotating the vector 180 degrees.

2.4.3 Some Additional Comments

You will note the the sum (or difference) of two vectors is always itself a vector. Remember, vectors can be represented as arrows because they have magnitude and direction. When you add two vectors together, you don’t lose the directionality element. If you did happen to lose the direction, you would essentially be saying that adding two vectors results in a scalar. That’s not okay.

For those of you who might attempt some trickery, let me clarify a specific circumstance. What if you have two vectors. $\vec{N}$ points north and has a magnitude of 2. $\vec{S}$ points south and has a magnitude of 2. Adding $\vec{N}+\vec{S}$ seems to result in a scalar answer: zero. Zero can’t have a direction, right? So it can’t possibly be a vector. I introduce to you $\vec{0}$ —the zero vector. For now, just accept it. We do physics here, not mathematics. The point is, no form of trickery can get around the fact that two vectors added or subtracted will always result in a vector quantity.

Chapter Summary

Vectors are an extremely important concept in physics. They provide a way to describe and understand physical quantities that have both magnitude and direction. By understanding vectors, we can better apply our mathematical toolbox to the world around us, thus enabling us to solve real-world problems that we could not do otherwise.

Enroll on Canvas

This course uses Canvas for homework assignments, quizzes, and exams. These assignments are open to everyone. Anyone is allowed to enroll in the Canvas course. In fact, this is highly encouraged as it will help you track your progress as you go through the course. Graded feedback will help you get an idea for what your grade would actually be in a Physics 1 college course. Use this link to enroll in the Canvas course .

Quiz 1 is also available and covers content from Chapters 1 and 2. Enroll in the Canvas course using the link above if you haven’t already. Then, click here to take the quiz.

Knowledge Check

Answer the quiz questions below.

A vector is defined as having:

Magnitude only

Close! These are referred to as scalars in physics.

Mass and Velocity

No, mass is a scalar while velocity is an example of a vector. However, neither defines a vector.

Direction only

Close! Direction is a key part of the definition, but it is not the only part.

Magnitude and Direction

Nice work! Vectors in physics describe quantities that have both a size and an associated direction, such as velocity, acceleration, and force.

Temperature is an example of a vector.

True

False

That’s right. Temperature is a scalar because it has no associated direction.

Why are vectors important in physics?

They allow us to calculate the weight of an object.

Incorrect, weight of an object can be calculated without the need for vectors.

They provide a way to describe time.

No, time is a scalar quantity and does not require vectors for its description.

They provide a powerful tool for describing and understanding motion and forces.

Correct! Vectors are essential for understanding and describing motion and forces in the world around us. For example, we can use vectors to calculate an object’s position or motion given its velocity and the forces acting on it.

In what application would vectors be used to describe the movement of air masses?

In sports analysis

No, sports analysis typically uses vectors to describe the velocity and acceleration of balls and athletes, not the movement of air masses.

In engineering

Incorrect. In engineering, vectors are often used to describe the forces acting on structures.

In weather forecasting

Correct! In weather forecasting, vectors are used to describe the movement of air masses, which can help meteorologists predict the direction and strength of storms.

How are vectors represented?

As numbers

Not exactly. While vectors do include numbers to represent magnitude, they also require a way to show direction.

As graphs

Not quite. While vectors can be shown on graphs, the vectors themselves are not graphs.

As arrows

Exactly! Vectors are best represented as arrows, which clearly depict both magnitude (through the length of the arrow) and direction (through the way the arrow is pointing).

In which of the following fields are vectors NOT typically used?

Engineering

Incorrect, vectors are used in engineering to describe the forces acting on structures.

Sports analysis

Incorrect, sports analysis uses vectors to describe the velocity and acceleration of balls and athletes.

Music production

That’s right! Music production typically does not require the use of vectors. Vectors are used in fields where understanding physical quantities that have both magnitude and direction is important.

Continue to Chapter 3: Vector Components

Back to Chapter 1: Introduction to the Metric System

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Chapter 2: Introduction to Vectors

2.1 Introduction

2.2 Vectors Help Us Describe and Understand Motion

2.3 Applications of Vectors