Vectors in 2D and 3D

Column vectors, i-j notation, and extending to three dimensions

SPEC 10.1 • VECTORS

Scalars vs Vectors

Scalar

A quantity with magnitude only

Examples: mass, temperature, speed, time, distance

Vector

A quantity with magnitude and direction

Examples: displacement, velocity, force, acceleration

Notation

Vectors are written as bold letters in print: a, b, v

In handwriting, use an underline: a̲, b̲, v̲

Directed Line Segments

A vector is shown as an arrow. The length shows magnitude, the arrow shows direction.

The 2D Coordinate System

In 2D, we use two perpendicular axes: the x-axis (horizontal) and y-axis (vertical).

Reading Components

The green vector goes 3 units right (positive x) and 2 units up (positive y).

Vectors in 2D: Column Notation

A vector in 2D can be written as a column vector:

Column Vector

Vector a = xywhere x = horizontal component, y = vertical component

Interpretation

The vector 32 means: move 3 units right and 2 units up

The vector −14 means: move 1 unit left and 4 units up

The Unit Vectors i and j

We define two special unit vectors (vectors with magnitude 1):

i

i = 10Unit vector in the positive x-direction

j

j = 01Unit vector in the positive y-direction

Component Form

Any 2D vector can be written as: a = xi + yj

Example: 3−2 = 3i − 2j

Worked Example01

Point A has coordinates (2, 3) and point B has coordinates (5, 7). Write the vector from A to B in column form.

1

To go from A to B, we need to find how far we move in each direction

2

Horizontal change: 5 − 2 = 3 (move 3 units right)

3

Vertical change: 7 − 3 = 4 (move 4 units up)

4

Write as a column vector: x-component on top, y-component below

AnswerVector from A to B = 34

Worked Example02

Convert 4 −3 to i-j notation.

1

The top component (4) is the coefficient of i

2

The bottom component (−3) is the coefficient of j

3

Write as: xi + yj = 4i + (−3)j

Answer4i − 3j

The 3D Coordinate System

In 3D, we add a third axis. All three axes are perpendicular (at right angles) to each other.

Visualising 3D

Think of standing at the origin: x goes right, y goes up, and z goes away from you (into the page).

Vectors in 3D: Column Notation

In three dimensions, we add a third component for the z-direction:

3D Column Vector

Vector a = xyzx = horizontal, y = vertical, z = depth (into page)

Example

The vector 2−15 means: move 2 right, 1 down, 5 away

The Unit Vector k

In 3D, we introduce a third unit vector:

i

100x-direction

j

010y-direction

k

001z-direction

3D Component Form

Any 3D vector can be written as: a = xi + yj + zk

Example: 2−15 = 2i − j + 5k

Remember

The order is always i, j, k corresponding to x, y, z components.

Worked Example03

Convert 2 i − 5j + 3k to column notation.

1

The coefficient of i is the x-component = 2

2

The coefficient of j is the y-component = −5

3

The coefficient of k is the z-component = 3

4

Stack the components vertically in order: x, y, z

Answer2−53

Worked Example04

Point P has coordinates (1, 2, 4) and point Q has coordinates (3, −1, 6). Write the vector from P to Q in i-j-k form.

1

Find the change in each component (end − start):

2

x-component: 3 − 1 = 2

3

y-component: −1 − 2 = −3

4

z-component: 6 − 4 = 2

5

Write in column form first: 2−32

6

Convert to i-j-k notation

AnswerVector from P to Q = 2i − 3j + 2k

Equal Vectors

Definition

Two vectors are equal if and only if all corresponding components are equal.

In 2D

ab = cd means a = c and b = d

In 3D

abc = def means a = d, b = e, and c = f

Important

Position doesn't matter for equality — only magnitude and direction. Two vectors in different positions can still be equal.

Worked Example05

Given a = 2p−1 and b = q3r, find the values of p, q, and r if a = b.

1

For vectors to be equal, all corresponding components must be equal

2

Comparing x-components (top row): 2 = q, so q = 2

3

Comparing y-components (middle row): p = 3, so p = 3

4

Comparing z-components (bottom row): −1 = r, so r = −1

Answerp = 3, q = 2, r = −1

Worked Example06

Sketch the vector 3 i − 2j on a coordinate grid, starting from the origin.

1

Convert to column form: 3i − 2j = 3−2

2

Starting from the origin (0, 0)

3

Move 3 units in the positive x-direction (right)

4

Move 2 units in the negative y-direction (down)

5

End point is at (3, −2)

AnswerDraw an arrow from (0, 0) to (3, −2)

Worked Example07

Point A has coordinates (1, 3, −2) and point B has coordinates (4, −1, 5). Find the vector from A to B, and the vector from B to A, giving your answers in both column and i-j-k notation.

1

Vector from A to B: Calculate end − start for each component

2

x: 4 − 1 = 3, y: −1 − 3 = −4, z: 5 − (−2) = 7

3

Vector A to B = 3−47 = 3i − 4j + 7k

4

Vector from B to A: This is the opposite direction, so negate each component

5

Vector B to A = −34−7 = −3i + 4j − 7k

Key insight: Vector from B to A = −(Vector from A to B)

Practice

Question 1: Convert −2i + 4j to column notation.

The coefficient of i is −2 (x-component)

The coefficient of j is 4 (y-component)

Answer−24

Practice

Question 2: Point M has coordinates (−1, 2, 3) and point N has coordinates (4, 0, −2). Write the vector from M to N in column form.

Find the change in each component (end − start):

x-component: 4 − (−1) = 4 + 1 = 5

y-component: 0 − 2 = −2

z-component: −2 − 3 = −5

Answer5−2−5

Practice

Question 3: The vector from point P(2, 5, −1) to point Q is 3i − j + 2k. Find the coordinates of Q.

The vector from P to Q tells us how to get from P to Q.

Vector = 3−12 means: go 3 in x, go −1 in y, go 2 in z

Starting at P(2, 5, −1):

x-coordinate of Q: 2 + 3 = 5

y-coordinate of Q: 5 + (−1) = 4

z-coordinate of Q: −1 + 2 = 1

AnswerQ = (5, 4, 1)

Summary

Key Concepts

Notation2D3D

Column Vectorx yx y z

Component Formx i + y jx i + y j + z k

Unit Vectorsi , ji , j , k

Key Exam Skills

Convert between column notation and i-j-k notation

Find the vector between two points: subtract coordinates (end − start)

Vector from B to A = −(Vector from A to B)

Use equal vectors to find unknown components