Vectors and Scalars in Physics

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2.1Vectors and Scalars

In this Physics tutorial, you will learn:

  • What are vectors? What are scalars?
  • How do vectors and scalars in Physics differ from those in Mathematics?
  • How can we express a vector quantity in components?
  • How to find the magnitude of a vector quantity when the components are given?
  • How to find the angle formed by a vector with a certain direction?

Introduction

Let's consider the following scenarios to understand the point.

  1. Your cousin has just arrived in your city. She calls you and asks to come and pick her with your car. When you ask her for the location she replies: "I am 2 km away from the city hall."
  2. Your father tells you to pull the dining table, as he wants to make place to a plant pot.
  3. You are lost in ocean during a trip with your ship. You don't have any mobile phone, radio or other tools of communication but only a map and a compass.
  4. You are told to kick a ball in a given direction but you are not able to see the person who told you to do so.

Are the information provided above complete? Do you need to know any other thing in each of the above scenarios?

Vectors and Scalars

Let's try to understand what is wrong with the situations described above. Thus,

  1. The information is incomplete, as your cousin didn't provide any direction. You cannot hang around the city to find her, despite she has informed you that she is at 2 km away from the city hall. Indeed, she may be anywhere in the circle whose centre is at the city hall and whose radius is 2 km long.
  2. The information is incomplete. Your father didn't tell you the direction in which you should pull the table. There are many possible directions to pull the table.
  3. The first thing to do in this case is trying to find the directions using the map and the compass. Then you can sail in the correct direction.
  4. In this case, you need to know whether you have to kick the ball gently or strongly. The person may be near or far away, so you have to know how to kick the ball despite you already know the direction.

In all these examples, there was some missing information. It was missing either the magnitude or the direction of the action you had to do. Therefore, it is quite impossible for you doing what you were told to.

The concept of vector helps you to understand the point. A vector (in mathematics) is a quantity that has both a magnitude (numerical value or size) and a direction. If one of them is missing, the information is incomplete.

Geometrically, a vector is represented through an arrow. The tip shows the direction and the ending point, while the toe shows the starting point of the vector. Symbolically, the vector is denoted in two possible ways:

  1. by two uppercase letters where the first letter shows the starting point and the second one the ending point of the vector, or:
  2. by a single lowercase letter at middle of the vector.

Also, a small arrow must be placed above the letter/s representing the vector to distinguish it from a segment. Look at the figure below.

Physics Tutorials: This image shows a long arrow joing points A to B with the letter v underneath (the v has a small arrow above indicating it is a vector). The arrow is followed by an equal sign and the letters AB with an arrow above to indicate a vector followed by the word

On the other hand, a quantity which has only magnitude (does not involve direction), is known as "scalar". For example, real numbers are scalars. You simply need to know their numerical value to have a complete information regarding the quantity involved. For example, we know that 5 > 3 because we compare the magnitudes of these two numbers (scalars).

Vectors and scalars in Physics

All quantities in Physics are either vector or scalar. For example, Force is a vector quantity as it involves direction. As discussed in the "Introduction" section, it is not sufficient to know only the magnitude or only the direction when trying to move something by exerting a force. We must know both the magnitude and direction of force to understand what to do exactly. On the other hand, Temperature is a scalar quantity because it doesn't involve any direction. It doesn't make sense if we say "the temperature today is 30°C due North". We simply say, "the temperature today is 30°C".

However, quantities in Physics require more info compared to those in Mathematics. If in Mathematics a vector quantity is fully known when two clues: magnitude and direction are given, in Physics we must also know the unit and the application point besides the two abovementioned clues. Let's explain this point through an example. (We will mention here the unit of force, Newton, whose symbol is [N], to illustrate the example. The meaning of newton as a unit of force will be discussed later, in the Section 4).

Example: You are told to move the object shown in the figure. This is an upper view of the object.

Physics Tutorials: This image shows a square with rounded edges, outside the square, the letters N, E, S, W are displayed with N at the top and the remaining letters displayed in clockwise order, indicating North, East, South and West

It is not sufficient if you are told "Pull the object due East". In this case, there is insufficient information, as you are not told the magnitude of the force to be used (how many newton of force must be used).

Also, it is not sufficient if you are told "Pull the object by 50 N" as no direction is mentioned.

Finally, it is not sufficient if you are told "Pull the object by 50 N due East" because no application point is provided. The person who told you to pull the object must also tell whether the force must be used at centre or at the edge of the object as the outcome will be different.

Thus, if you pull the object at its centre, it will move like this

Physics Tutorials: This image shows a rounded square with an arrow indicating a direction of movement. Their is a silhoutte of the same object further to the right of the original to indicate the objects final position after being pulled

But if you pull the object by picking it at the edge, the result will be as shown below:

Physics Tutorials: This image shows a rounded suqare similar to the previous image though, the silhoutted object has now be turned 45 degrees clockwise in addition to being moved to illustrate rotation motion

In the first case, the object will make only translational motion (parallel shift) while in the second case there will be a combination of translational and rotational motion as the object first starts rotating and then it moves linearly.

The differences between vector and scalar quantities in Maths and Physics are summarized in the table below.

Differences between vector and scalar quantities in Maths and Physic
Property Scalars in MathsVectors in MathsScalars in PhysicsVectors in Physics
Magnitude
Direction××
Unit××
Application point×××

Components of a vector

It is not always the case that the direction of a vector coincides with one of the main directions we use as basic ones (up, down, left, right, or North, South, West, East). Vectors can also lie in such directions that are combinations of these four basic ones such as the vector shown in the figure below.

Physics Tutorials: This image shows

The direction of the vector AB is neither due East, nor due North but it is a combination of these two basic directions. Therefore, we can provide info about the horizontal and vertical shift of the point B in relation to the point A if the vector AB is not visually shown. In this way, you will enable the listener drawing the vector if he wants to. For example, if you say: "the point B is 4 units on the right and 3 units above the point A", your partner will easily draw the vector AB as shown below.

Physics Tutorials: This image shows

In this way, the vector AB splits into two components: the horizontal (x - component) and the vertical (y - component). Look at the figure.

Physics Tutorials: This image shows

Magnitude of a vector. How to find the magnitude of a vector quantity when the components are given?

It is not easy to find the length (magnitude) of the vector AB when it is does not lie according one of the main directions, especially when you don't have any measuring tool but the only info you have are the horizontal and vertical units.

From the figure above, it is obvious that when splitting a vector into components, a right triangle is obtained. These components, namely ABx⃗ and ABy are those which form the right angle (legs) and the vector AB itself is the hypotenuse of the right triangle. From the Pythagorean Theorem, we know that

AB2 = AB2x + AB2y

Therefore, the magnitude (size) of the vector AB is

|AB| = √AB2x + AB2y

(It is not necessary to write the vector symbol (→) above the components. Also, the symbol | | stands for the magnitude of the vector. It mathematically represents the absolute value of the quantity it contains. This is because the vector can lie to the negative direction but its magnitude is always positive as it represents the vector's length.)

Let's explain this point through an example.

Example

The horizontal and vertical components of a force are Fx = 120 N and Fy = 160 N respectively. What is the magnitude of the force F?

Solution

In fact, the components of a vector (in the specific case, we have a Force vector) have their starting point at the starting point of the original vector itself. However, for practical purpose, we can shift the vertical component Fy in a parallel way in order to close the triangle.

Physics Tutorials: This image shows

From the Pythagorean Theorem we have

|F|2 = F2x + F2y
|F|2 = 1202 + 1602
= 14400 + 25600
= 40000

Therefore, the magnitude of the force F is

|F| = √40000
= 200N

How to find the angle formed by a vector with a certain direction?

From Trigonometry, we know that in a right triangle with legs a and b respectively and hypotenuse c as in the figure below, we have

Physics Tutorials: This image shows
cos θ = Adjacent side/Hypotenuse = a/c
vsin θ = Opposite side/Hypotenuse = b/c
cot θ = Adjacent side/Opposite side = a/b
tan θ = Opposite side/Adjacent side = b/a

where Θ is the angle formed by the adjacent side and hypotenuse. These rules can also apply in vectors. They help us find the direction of a vector quantity. We simply replace a with vx, b with vy and c with the vector v, whatever it may represent.

Example 2

An airplane moves 44 km due North and then 76 km due East. Calculate:

  1. The total displacement of the airplane
  2. The angle formed by its displacement vector to the West - East direction.

(Displacement is a vector quantity)

Solution 2

The airplane's flight is graphically described in the figure below.

Physics Tutorials: This image shows

It is obvious the angles Θ are both equal as they are alternate interior angles. Therefore, we can limit our study by considering only the right triangle.

The two given values (44 km and 76 km) are the components of the displacement vector as they are perpendicular to each other and furthermore, they lie according the two basic directions. Thus, we have

(Total displacement)2 = (Displaement due East)2 + (Displaement due North)2

(Despite the displacement due North has occurred first, we can switch the position of the addends due to the commutative property of addition)

Substituting the values, we obtain

(Total displacement)2 = (44km)2 + (76km)2
= (1936 + 5776) km2
= 7712 km2

Here km2 does not stand for any surface area but only because we raised the displacements at power two).

Therefore, the total displacement will be

Total displacement = √7712 km2
= 87.8 km

This value represents the shortest path from the initial to the final position of the airplane, as shown by the grey vector

We can use the trigonometric equations to find the required direction. The direction is considered as known if we find the angle Θ formed by the displacement vector to the West-East direction (here it is similar to the x-axis). Thus, when considering the triangle formed in the graph, we obtain

tan θ = Opposite side/Adjacent side
= Displacement due North/Displacement due East
= 44km/76 km= 11/19

Therefore, the angle Θ is

Θ = tan-1 (11/19) = 30.07°

Hence, the airplane is 87.8 km away from the starting position and its displacement vector forms an angle of 30.07° to the West-East direction.

This information is particularly useful when the pilot provides his location during the communication with the air-traffic control centre during the flight.

We hope you found this tutorial useful, if you did. Please take the time to rate this tutorial and/or share on your favourite social network. In our next tutorial, we explore Addition and Subtraction of Vectors.

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