Vector Operations

A vector is a quantity that has both magnitude and direction.

Examples of vector quantities include:

• displacement
• velocity
• acceleration
• force
• momentum

Vectors can be added, subtracted, and multiplied by numbers called scalars. These operations allow us to combine several vectors into a single resultant vector.

For example, if two forces act on an object, vector addition can determine their combined effect.

Vector Notation

To keep the notation clear and compatible with Moodle, we will represent vectors using ordinary capital letters such as:

A, B, C, R

When working with components, a two-dimensional vector can be written:

A = (Aₓ, Aᵧ)

For example:

A = (3, 4)

means the vector has:

• horizontal component = 3
• vertical component = 4

The direction of each component is determined by its sign.

A common convention is:

• right → positive x
• left → negative x
• up → positive y
• down → negative y

The magnitude of a vector is represented using vertical bars:

|A| = magnitude of A

Scalars and Vectors

A scalar has magnitude but no direction.

Examples include:

• mass
• temperature
• time
• distance
• energy

A vector has both magnitude and direction.

For example:

5 m

is a scalar distance.

5 m east

is a vector displacement.

This distinction becomes especially important when quantities are combined.

Adding Vectors Graphically

Vectors can be added using the head-to-tail method.

Suppose we want to calculate:

A + B

To add them:

1. Draw vector A.
2. Place the tail of B at the head of A.
3. Draw the resultant from the tail of A to the head of B.

The resultant is:

R = A + B

Example: Adding Displacements

A person walks:

4 m east

and then:

3 m north

These two movements form perpendicular vectors.

The total distance travelled is:

4 + 3 = 7 m

But the displacement is the straight-line vector from the starting point to the finishing point.

Using Pythagoras:

R = √(4² + 3²)

R = √(16 + 9)

R = √25

R = 5 m

So the resultant displacement has magnitude 5 m.

Notice:

distance = 7 m

but:

displacement magnitude = 5 m

Direction of the Resultant

For the previous example:

horizontal component = 4 m

vertical component = 3 m

The direction can be calculated using:

tan θ = vertical component / horizontal component

Therefore:

tan θ = 3/4

θ = tan⁻¹(3/4)

θ ≈ 36.9°

The resultant displacement is:

5 m at 36.9° north of east

Adding Vectors Algebraically

If vectors are written using components, addition is straightforward.

Suppose:

A = (3, 5)

and:

B = (4, 2)

Add the x-components:

3 + 4 = 7

Add the y-components:

5 + 2 = 7

Therefore:

A + B = (7, 7)

The general rule is:

(a, b) + (c, d) = (a + c, b + d)

Worked Example

Let:

A = (6, 2)

B = (−3, 5)

Find:

A + B

Add the x-components:

6 + (−3) = 3

Add the y-components:

2 + 5 = 7

Therefore:

A + B = (3, 7)

Adding More Than Two Vectors

The same method works when several vectors are involved.

Suppose:

A = (4, 3)

B = (−2, 5)

C = (6, −4)

Find:

R = A + B + C

Add the x-components:

Rₓ = 4 − 2 + 6

Rₓ = 8

Add the y-components:

Rᵧ = 3 + 5 − 4

Rᵧ = 4

Therefore:

R = (8, 4)

This component method is particularly useful for multi-step vector problems.

Subtracting Vectors

Vector subtraction can be treated as adding the negative of a vector.

Therefore:

A − B = A + (−B)

The negative of a vector has:

• the same magnitude
• the opposite direction

If:

B = (4, 2)

then:

−B = (−4, −2)

Subtraction Using Components

Suppose:

A = (7, 5)

B = (2, 3)

Then:

A − B = (7 − 2, 5 − 3)

Therefore:

A − B = (5, 2)

The general rule is:

(a, b) − (c, d) = (a − c, b − d)

Worked Subtraction Example

Let:

A = (4, 8)

B = (−2, 3)

Find:

A − B

x-component:

4 − (−2) = 6

y-component:

8 − 3 = 5

Therefore:

A − B = (6, 5)

Be particularly careful when subtracting negative components.

Multiplying a Vector by a Scalar

A scalar is an ordinary number.

Multiplying a vector by a scalar changes its magnitude.

For example:

A = (3, 2)

Then:

2A = (6, 4)

Both components have been multiplied by 2.

The vector is now twice as long but points in the same direction.

Positive Scalar Multiplication

Suppose:

A = (2, 5)

Find:

3A

Multiply each component:

3A = (3 × 2, 3 × 5)

Therefore:

3A = (6, 15)

The magnitude becomes three times larger.

The direction remains unchanged.

Negative Scalar Multiplication

Suppose:

A = (4, 2)

Find:

−2A

Multiply both components:

−2A = (−8, −4)

Therefore:

−2A = (−8, −4)

The magnitude is doubled, but the direction is reversed.

In general:

• positive scalar → same direction
• negative scalar → opposite direction
• scalar greater than 1 → vector becomes longer
• scalar between 0 and 1 → vector becomes shorter

Finding the Magnitude of a Vector

For a two-dimensional vector:

A = (Aₓ, Aᵧ)

the magnitude is found using Pythagoras:

|A| = √(Aₓ² + Aᵧ²)

The vertical bars mean:

|A| = magnitude of A

They do not mean that A is being multiplied by anything.

Worked Example: Magnitude

Find the magnitude of:

A = (6, 8)

Use:

|A| = √(6² + 8²)

|A| = √(36 + 64)

|A| = √100

|A| = 10

So vector A has magnitude 10 units.

Resultant Vectors

A resultant vector is the single vector that has the same overall effect as two or more vectors combined.

For example:

R = A + B

If:

A = (5, 2)

B = (−1, 4)

then:

R = (5 − 1, 2 + 4)

R = (4, 6)

The magnitude is:

|R| = √(4² + 6²)

|R| = √52

|R| ≈ 7.21

Finding Direction from Components

Suppose:

R = (8, 6)

Magnitude:

|R| = √(8² + 6²)

|R| = √100

|R| = 10

Direction:

tan θ = 6/8

θ = tan⁻¹(0.75)

θ ≈ 36.9°

Because both components are positive, the vector points up and to the right.

Therefore, the resultant is:

10 units at 36.9° above the positive x-axis

Components from Magnitude and Direction

Sometimes we know a vector's magnitude and angle instead of its components.

If vector A has magnitude A and angle θ measured from the positive x-axis:

Aₓ = A cos θ

Aᵧ = A sin θ

This allows the vector to be converted into horizontal and vertical components.

Worked Example: Resolving a Vector

A force of 20 N acts at an angle of 30° above the horizontal.

Horizontal component:

Fₓ = 20 cos 30°

Fₓ ≈ 17.3 N

Vertical component:

Fᵧ = 20 sin 30°

Fᵧ = 10.0 N

Therefore, the force can be represented as:

F = (17.3, 10.0) N

This technique is called resolving a vector into components.

Why Components Are Useful

Suppose several forces act at different angles.

Adding the vectors directly from a diagram can become difficult.

Instead:

1. resolve each vector into x- and y-components
2. add all x-components
3. add all y-components
4. calculate the magnitude of the resultant
5. calculate its direction

This provides a systematic method for complicated vector problems.

Multi-Step Vector Problem

Three forces act on an object:

F₁ = (8, 3) N

F₂ = (−2, 5) N

F₃ = (4, −6) N

Find the resultant.

Add the x-components

Rₓ = 8 − 2 + 4

Rₓ = 10 N

Add the y-components

Rᵧ = 3 + 5 − 6

Rᵧ = 2 N

Therefore:

R = (10, 2) N

Find the magnitude

|R| = √(10² + 2²)

|R| = √104

|R| ≈ 10.2 N

Find the direction

tan θ = 2/10

θ = tan⁻¹(0.2)

θ ≈ 11.3°

Therefore, the resultant is approximately:

10.2 N at 11.3° above the positive x-axis

Multi-Step Problem with Scalar Multiplication

Let:

A = (2, 5)

B = (4, −1)

Find:

R = 3A − 2B

First calculate:

3A = (6, 15)

Then:

2B = (8, −2)

Now subtract:

R = (6, 15) − (8, −2)

R = (6 − 8, 15 − (−2))

Therefore:

R = (−2, 17)

Magnitude:

|R| = √[(−2)² + 17²]

|R| = √293

|R| ≈ 17.1

This example combines scalar multiplication, subtraction, and magnitude.

Vector Operations with Forces

Suppose an object experiences four forces:

• 12 N east
• 5 N west
• 8 N north
• 2 N south

Choose:

east = positive x

north = positive y

Horizontal resultant:

Rₓ = 12 − 5

Rₓ = 7 N

Vertical resultant:

Rᵧ = 8 − 2

Rᵧ = 6 N

Therefore:

R = (7, 6) N

Magnitude:

|R| = √(7² + 6²)

|R| = √85

|R| ≈ 9.22 N

Direction:

tan θ = 6/7

θ ≈ 40.6°

Therefore:

R ≈ 9.22 N at 40.6° north of east

Vector Operations with Velocity

Suppose an aircraft moves at:

200 km/h east

while the wind moves at:

50 km/h north

Represent the vectors as:

Aircraft velocity:

V₁ = (200, 0)

Wind velocity:

V₂ = (0, 50)

Resultant:

V = V₁ + V₂

V = (200, 50)

Magnitude:

|V| = √(200² + 50²)

|V| = √42500

|V| ≈ 206 km/h

The actual motion relative to the ground is therefore different from the aircraft's velocity relative to the air.

Vector Operations with Displacement

Suppose a hiker travels:

• 5 km east
• 3 km north
• 2 km west
• 1 km south

Horizontal displacement:

Rₓ = 5 − 2

Rₓ = 3 km east

Vertical displacement:

Rᵧ = 3 − 1

Rᵧ = 2 km north

Resultant:

R = (3, 2) km

Magnitude:

|R| = √(3² + 2²)

|R| = √13

|R| ≈ 3.61 km

The hiker travelled a total distance of:

5 + 3 + 2 + 1 = 11 km

but the magnitude of the final displacement is only:

3.61 km

This demonstrates again that distance and displacement are different quantities.

Three-Dimensional Vectors

Vectors can also have three components.

For example:

A = (3, 4, 12)

The magnitude is:

|A| = √(3² + 4² + 12²)

|A| = √(9 + 16 + 144)

|A| = √169

|A| = 13

The same vector operations apply:

(a, b, c) + (d, e, f) = (a + d, b + e, c + f)

Three-dimensional vectors are important in physics, engineering, aviation, and computer graphics.

Graphical vs. Algebraic Methods

Graphical Method

Advantages:

• shows direction visually
• helps develop understanding
• useful for estimating resultants

Limitations:

• depends on accurate drawing
• difficult with many vectors
• results may only be approximate

Algebraic Method

Advantages:

• precise
• efficient
• works well with many vectors
• suitable for multi-step problems

Limitations:

• requires careful signs and calculations
• direction may be less visually obvious

For complex problems, the algebraic component method is usually more reliable.

A General Strategy for Multi-Step Vector Problems

When solving a complicated vector problem:

Identify the vectors.

Determine their magnitudes and directions.

Choose coordinate directions.

Usually:

right = +x

left = −x

up = +y

down = −y

Resolve angled vectors.

Use:

Vₓ = V cos θ

Vᵧ = V sin θ

Perform the required operations.

Add, subtract, or multiply components as needed.

Find the resultant components.

R = (Rₓ, Rᵧ)

Calculate magnitude.

|R| = √(Rₓ² + Rᵧ²)

Calculate direction if required.

tan θ = Rᵧ/Rₓ

Finally, check the signs of Rₓ and Rᵧ so that the direction makes physical sense.

Common Mistakes

Adding magnitudes instead of vectors

If vectors point in different directions, their magnitudes cannot simply be added.

Ignoring negative components

Leftward and downward components normally require negative signs.

Subtracting only the magnitudes

Vector subtraction means adding the opposite vector or subtracting corresponding components.

Forgetting to multiply every component by a scalar

If:

A = (3, 5)

then:

2A = (6, 10)

not (6, 5).

Using Pythagoras for non-perpendicular vectors without resolving them

Pythagoras applies directly when the components or vectors are perpendicular.

Confusing magnitude with a component

For:

R = (3, 4)

the magnitude is:

|R| = 5

not 3 + 4 = 7.

Using the wrong angle

Always identify which axis the angle is measured from before choosing sine and cosine.

Ignoring the quadrant

A calculator may give a reference angle that does not completely describe the vector's direction.

Key Terms

Vector — a quantity with both magnitude and direction.

Scalar — a quantity with magnitude but no direction.

Magnitude — the size or length of a vector.

Direction — the orientation in which a vector points.

Component — the part of a vector acting along a particular axis.

Resultant vector — the single vector representing the combined effect of several vectors.

Vector addition — combining vectors to produce a resultant.

Vector subtraction — subtracting one vector from another, which can also be understood as adding the negative vector.

Negative vector — a vector with the same magnitude as another vector but the opposite direction.

Scalar multiplication — multiplying every component of a vector by a scalar.

Head-to-tail method — a graphical technique in which vectors are placed consecutively to determine their resultant.

Resolving a vector — separating a vector into perpendicular components.

x-component — the horizontal component of a vector.

y-component — the vertical component of a vector.

Key Takeaways

• Vectors have both magnitude and direction.
• Vectors can be added graphically using the head-to-tail method.
• Algebraically, vectors are added by adding corresponding components.
• Vector subtraction can be treated as addition of the negative vector.
• Multiplying a vector by a scalar changes its magnitude and may reverse its direction.
• For A = (x, y), its magnitude is |A| = √(x² + y²).
• The notation |A| means the magnitude of vector A.
• Angled vectors can be resolved using Vₓ = V cos θ and Vᵧ = V sin θ.
• Multiple vectors can be combined by adding all x-components and all y-components.
• A resultant vector represents the overall effect of several vectors.
• Component methods are especially useful for multi-step problems involving forces, velocities, and displacements.
• Signs and directions are just as important as numerical values when working with vectors.

Check Your Understanding

1. Add:

   A = (3, 4)

   B = (5, 2)

2. Find A − B if:

   A = (8, 6)

   B = (3, 9)

3. Given:

   A = (4, −2)

   calculate 3A.

4. Given:

   B = (5, 3)

   calculate −2B.

5. Find the magnitude of:

   R = (5, 12)

6. Two perpendicular forces of 9 N east and 12 N north act on an object. Find the magnitude of the resultant.

7. Find the magnitude and direction of:

   R = (12, 5)

8. A 30 N force acts at 40° above the horizontal. Determine its horizontal and vertical components.

9. Given:

   A = (3, 5)

   B = (−2, 4)

   C = (6, −3)

   Find A + B + C and determine its magnitude.

10. Given:

A = (2, 4)

B = (5, −1)

Calculate 3A − 2B and determine the magnitude of the resultant.

11. A hiker walks 8 km east, 6 km north, 3 km west, and 2 km south. Determine the magnitude and direction of the final displacement.
12. Explain why simply adding the magnitudes of several vectors does not usually give the magnitude of their resultant.