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.

What Are Vector Components?

A vector has both magnitude and direction.

Sometimes it is easier to work with a vector by splitting it into two simpler vectors called components.

For a two-dimensional vector, we normally use:

• A horizontal component along the x-axis.
• A vertical component along the y-axis.

Together, these two components have the same overall effect as the original vector.

For example, a force acting diagonally upward can be separated into:

• A force acting horizontally.
• A force acting vertically.

This process is called resolving a vector into components.

 

 

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Understanding the Vector Triangle

When a vector is resolved into horizontal and vertical components, the three vectors form a right-angled triangle.

Suppose a vector V

has magnitude V and makes an angle θ above the horizontal.

We can label:

• V = magnitude of the original vector
• V_x​ = horizontal component
• V_y​ = vertical component
• θ = angle measured from the horizontal

The original vector is the hypotenuse of the triangle.

The horizontal and vertical components are the other two sides.

This allows us to use trigonometry to calculate the components.

Resolving a Vector Using Trigonometry

Remember the trigonometric relationships:

\( sin \theta = \frac{oposite}{hypotenuse} , \ cos \theta = \frac{adjacent}{hypotenuse} \)​

If the angle is measured from the horizontal, the horizontal component is adjacent to the angle and the vertical component is opposite.

Therefore:

V_x​ = Vcosθ​

and

V_y​ = Vsinθ​

These are two of the most important equations when working with vectors.

x = rcosθ, y = rsinθ

Remember

If the angle is measured from the horizontal:

Horizontal → cosine

Vertical → sine

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Worked Example 1: Finding Components

A force of 50 N acts at an angle of 30∘ above the horizontal.

Find the horizontal and vertical components.

Step 1: Identify the information

V = 50 N 

 

θ = 30^o

 

Step 2: Find the horizontal component

V_x​ = Vcosθ

 

Vx​ = 50cos30^o

 

V_x​ ≈ 43.3 N

 

Step 3: Find the vertical component

V_y​ = Vsinθ

 

V_y​ = 50sin30^o

 

V_y​ = 25.0 N

Therefore, the vector has components:

V_x​ = 43.3 N​ V_y ​= 25.0 N​

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Component Notation

Vectors can also be written using component notation.

For example:

\( \vec{v} \) = ⟨6,8⟩

This means:

V_x​ = 6 and V_y ​ = 8

The first number represents the horizontal component, and the second represents the vertical component.

Another common notation is:

\( \vec{v} \) = 6i + 8j

where:

• i represents the positive x-direction.
• j represents the positive y-direction.

Both forms describe the same vector.

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Positive and Negative Components

The signs of the components tell us the direction.

For the horizontal component:

• Positive x → right
• Negative x → left

For the vertical component:

• Positive y → up
• Negative y → down

For example:

\( \vec{v} \) = ⟨−4, 7⟩

means the vector points:

• 4 units to the left
• 7 units up

Therefore, the vector points generally up and to the left.

 

 

 

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Reconstructing a Vector

Sometimes we know the horizontal and vertical components and need to determine the original vector.

Suppose:

V_x​ = 6 m/s

and

V_y ​ = 8  m/s

Because the components form a right-angled triangle, we can use the Pythagorean theorem:

\( V^2 = V_x^2 + V_y^2 \)​

Therefore:

\( V = \sqrt[]{V_x^2 + V_y^2} \)

​​

Substituting the values:

\( V = \sqrt[]{6^2 + 8^2} \)

 

\( V = \sqrt[]{36 + 64} = \sqrt[]{100} \)​

 

​V = 10 m/s​

The magnitude of the original vector is 10 m/s.

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Finding the Direction

We can also use the components to determine the vector's direction.

Using:

\( tan \theta = \frac{V_y}{V_x} \)

​​

we can calculate:

\( \theta = tan^{-1}( \frac{V_y}{V_x} ) \)

For our vector:

V_x = 6;  V_y ​= 8

Therefore:

\( \theta = tan^{-1}( \frac{8}{6}) \approx 53.1^o \)

The complete vector is therefore:

10 m/s at 53.1^o above the horizontal​

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Resolving Forces

Vector components are particularly important when working with forces.

Imagine a person pulling a box using a rope at an angle.

 

 

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Suppose the person pulls with:

F = 100 N at: 30^o above the horizontal.

The horizontal component is:

F_x​ = 100cos30^o ​≈ 86.6 N

The vertical component is:

F_y​ = 100sin30^o  ​= 50 N

The horizontal component helps move the box forward, while the vertical component pulls slightly upward on the box.

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Vector Components in Projectile Motion

Vector components are also important when studying objects moving through the air.

Imagine a football kicked at an angle.

Its initial velocity can be separated into:

• Horizontal velocity
• Vertical velocity

 

 

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Suppose a ball is launched at: 20 m/s at an angle of: 40^o

The horizontal velocity is:

v_x​ = 20cos40^o ≈ 15.3 m/s

The vertical velocity is:

v_y​ = 20sin40^o ​≈ 12.9 m/s

These components can then be analysed separately when studying the ball's motion.

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Components in Navigation

Vector components are also useful in navigation.

Suppose a boat travels northeast.

Its velocity can be separated into:

• An eastward component.
• A northward component.

Similarly, wind acting on an aircraft can be separated into horizontal directions so that pilots can calculate the aircraft's actual motion.

 

 

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A Useful Problem-Solving Method

When solving vector-component problems:

1. Draw the vector

Sketch the vector and its direction.

2. Draw the components

Create horizontal and vertical arrows to form a right triangle.

3. Identify the angle

Check carefully whether the angle is measured from the horizontal or vertical.

4. Choose the correct trigonometric relationship

If the angle is measured from the horizontal:

V_x​ = Vcosθ

 

V_y ​= Vsinθ

 

5. Calculate

Substitute the values and include the correct units.

6. Check the signs

Make sure the signs of the components match their directions.

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Be Careful: Where Is the Angle?

You should not automatically assume that cosine always gives the horizontal component.

The formulas

V_x​ = Vcosθ and V_y​ = Vsinθ work when θ is measured from the horizontal.

If the angle is measured from the vertical, the relationships switch.

For example:

 

 

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The safest approach is to identify:

• Hypotenuse → original vector
• Adjacent → use cosine
• Opposite → use sine

Then use SOH CAH TOA rather than simply memorizing which component uses sine or cosine.

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Worked Example 2: Practical Problem

A rescue helicopter travels at 80 m/s at an angle of 25∘ north of east.

Horizontal component

v_x​ = 80cos25^o ​≈ 72.5 m/s

 

Vertical component

v_y​ = 80sin25^o ≈ 33.8 m/s

Therefore, the helicopter's velocity can be written as:

\( \vec{v} \) = ⟨72.5, 33.8⟩ m/s​

This tells us that the helicopter is travelling approximately:

• 72.5 m/s east
• 33.8 m/s north

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Did You Know?

Engineers and computer programmers often work with vectors by using their components rather than their magnitude and direction.

Computer games, flight simulators, GPS systems, robotics, and 3D animation all use vector components to calculate how objects move through space.

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Key Terms

Vector component – One part of a vector acting in a particular direction.

Horizontal component – The part of a vector acting along the x-axis.

Vertical component – The part of a vector acting along the y-axis.

Resolve – To separate a vector into its components.

Reconstruct – To combine components to determine the original vector.

Magnitude – The size of a vector.

Component notation – A way of describing a vector using its horizontal and vertical components, such as ⟨6,8⟩.

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Key Takeaways

• A vector can be resolved into horizontal and vertical components.
• The components form a right-angled triangle with the original vector.
• If the angle is measured from the horizontal: Vx​ = Vcosθ​ Vy ​= Vsinθ​
• A vector can be reconstructed using: \( V = \sqrt[]{V_x^2 + V_y^2} \)​​​
• Its direction can be found using: \( \theta = tan^{-1}( \frac{V_y}{V_x}) \)​
• Positive and negative component values indicate different directions.
• Always check where the angle is measured from before choosing sine or cosine.
• Vector components are used extensively in forces, projectile motion, navigation, engineering, and computer simulations.