Fundamentals of Vectors

Scalars and Vectors

In mathematics and physics, quantities can often be divided into two groups: scalars and vectors.

A scalar quantity has magnitude only.

Magnitude simply means the size or amount of a quantity.

For example:

• A temperature of 25^oC
• A mass of 5 kg
• A time of 10 s
• A speed of 60 km/h

None of these quantities needs a direction.

A vector quantity has both:

1. Magnitude
2. Direction

For example:

A car travelling at 60 km/h east

The magnitude is 60 km/h, while the direction is east.

Because both magnitude and direction are given, this is a vector quantity.

Scalar vs Vector Quantities

Compare these two statements:

The cyclist travels at 15 m/s.

This gives only a magnitude, so it describes speed, which is a scalar.

The cyclist travels at 15 m/s north.

This gives both magnitude and direction, so it describes velocity, which is a vector.

Key Idea

Scalar = magnitude only

Vector = magnitude + direction

Magnitude and Direction

Every vector contains two pieces of information.

Consider:

20 m east

The magnitude is:

20 m

The direction is:

east

Now consider a force:

50 N downward

The magnitude is 50 N, and the direction is downward.

Changing either the magnitude or the direction changes the vector.

For example:

10 N east

and

10 N west

have the same magnitude but are different vectors because they point in different directions.

Representing Vectors with Arrows

Vectors can be represented graphically using arrows.

 

 

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The arrow tells us two things:

• The length of the arrow represents the magnitude.
• The arrowhead represents the direction.

A longer arrow represents a larger magnitude.

For example, if we use the scale:

1 cm = 10 N

then a force of 30 N would be represented by an arrow that is 3 cm long.

Drawing Vectors to Scale

When vectors are drawn accurately, we can use a scale.

Suppose a person walks:

40 m east

We could choose:

1 cm = 10 m

The vector should therefore be drawn as a 4 cm arrow pointing east.

Similarly, a displacement of:

20 m north

would be represented by a 2 cm arrow pointing north.

 

 

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Drawing vectors to scale becomes especially useful when we later need to add or compare vectors.

Vector Notation

Vectors can also be represented symbolically.

A common method is to place an arrow above a letter: \( \vec{v} \)

This means the vector v.

For example: \( \vec{F} \) can represent a force vector.

\( \vec{v} \) can represent a velocity vector.

\( \vec{a} \) can represent an acceleration vector.

Another common notation uses bold letters, such as: v

Both notations tell us that the quantity has magnitude and direction.

Magnitude of a Vector

Sometimes we want to refer only to the magnitude of a vector.

If the vector is: \( \vec{v} \)

its magnitude can be written as: |\( \vec{v} \)|

For example, suppose:

\( \vec{v} \) = 15 m/s east

Then: |\( \vec{v} \)| = 15 m/s

Notice that the magnitude does not include the direction.

Describing Direction

There are several ways to describe the direction of a vector.

Compass Directions

Vectors can point:

• North
• South
• East
• West

or combinations such as:

• Northeast
• Northwest
• Southeast
• Southwest

Angles

Directions can also be described using angles.

For example:

20 m at 30^o above the horizontal

This tells us both the magnitude and the precise direction of the vector.

 

 

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Distance and Displacement

One important example of the difference between scalars and vectors is distance and displacement.

Distance is the total length of the path travelled.

Distance is a scalar because it has magnitude only.

Displacement is the change in position from the starting point to the finishing point.

Displacement is a vector because it has both magnitude and direction.

Imagine that a person walks:

5 m east

and then:

5 m west

The total distance travelled is:

5 + 5 = 10 m

However, the person finishes where they started.

Therefore:

displacement = 0 m

 

 

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Speed and Velocity

Another important pair is speed and velocity.

Speed describes how fast an object moves.

For example: 20 m/s

Speed is a scalar because no direction is required.

Velocity describes how fast an object moves and in what direction.

For example: 20 m/s north

Velocity is therefore a vector.

This means that an object's velocity can change even if its speed remains constant.

For example, a car travelling around a circular track at a constant speed is continuously changing its direction.

Therefore, its velocity is changing.

Forces Are Vectors

Force is another important vector quantity.

Suppose two students push a box.

One student pushes:

50 N east

while another pushes:

50 N west

The forces have the same magnitude but opposite directions.

Because force is a vector, the directions are extremely important.

 

 

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If the forces are equal and opposite, they balance each other.

The resultant force is:

0 N

Vectors in Real-World Situations

Vectors are used whenever both size and direction are important.

Navigation

An aircraft might travel: 800 km/h northwest

Both speed and direction are needed to determine where the aircraft will travel.

Weather

Wind may be described as: 30 km/h east

Meteorologists need both wind speed and wind direction.

Forces

Engineers must consider the magnitude and direction of forces acting on buildings, bridges, vehicles, and other structures.

Sports

The motion of a football, basketball, or golf ball can be described using velocity vectors.

 

 

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Worked Example

A boat travels at: 12 m/s north

Step 1: Identify the magnitude

12 m/s

Step 2: Identify the direction

north

Step 3: Classify the quantity

Because the quantity has both magnitude and direction, it is a vector.

We could represent it with an arrow pointing north.

If the scale were:

1 cm = 4 m/s

the arrow would need to be:

\( \frac{12}{4} \) = 3 cm long.

Therefore, we would draw a 3 cm arrow pointing north.

Did You Know?

Pilots and ship captains must constantly work with vectors.

An aircraft may point in one direction while wind pushes it in another direction. Pilots must consider both the aircraft's velocity and the wind velocity to determine the aircraft's actual motion over the ground.

This is one reason why understanding vectors is so important in navigation.

Key Terms

Scalar – A quantity that has magnitude only.

Vector – A quantity that has both magnitude and direction.

Magnitude – The size or amount of a quantity.

Direction – The orientation in which a vector points.

Vector notation – Symbols such as \( \vec{v} \) or v used to represent vectors.

Distance – The total length of a path travelled; a scalar quantity.

Displacement – The change in position from the starting point to the finishing point; a vector quantity.

Speed – The rate of movement without direction; a scalar quantity.

Velocity – Speed in a particular direction; a vector quantity.

Key Takeaways

• A scalar has magnitude only.
• A vector has both magnitude and direction.
• Distance, speed, mass, time, and temperature are examples of scalars.
• Displacement, velocity, acceleration, force, and momentum are examples of vectors.
• Vectors can be represented using arrows.
• The length of a vector arrow represents its magnitude.
• The arrowhead shows its direction.
• Vectors can be written using notation such as v.
• The magnitude of a vector can be written as ∣v∣.
• Vectors are important in physics, engineering, navigation, weather, and many other real-world situations.