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This lesson covers two-dimensional vectors — their representation, magnitude, direction, and operations — as required by the Edexcel A-Level Mathematics specification (9MA0). You need to understand column vector notation, calculate magnitudes, find unit vectors, and perform addition, subtraction, and scalar multiplication.
A vector is a quantity that has both magnitude (size) and direction. This distinguishes it from a scalar, which has only magnitude.
| Type | Examples |
|---|---|
| Scalar | Speed, mass, temperature, distance |
| Vector | Velocity, force, displacement, acceleration |
Vectors are typically written in bold (e.g., a) or with an arrow above (e.g., a with arrow). In handwriting, you underline the letter: a̲.
A 2D vector can be written as a column vector:
a = (x, y) written vertically as a column
where x is the horizontal component and y is the vertical component.
Example: The vector from point A(1, 3) to point B(4, 7) is:
AB = (4 - 1, 7 - 3) = (3, 4) as a column vector
This means: move 3 units in the positive x-direction and 4 units in the positive y-direction.
Vectors can also be written using the unit vectors i and j:
So the vector (3, 4) can be written as 3i + 4j.
Example: v = (5, -2) = 5i - 2j
Exam Tip: Both column vector notation and i/j notation are used at A-Level. Be comfortable switching between them.
The magnitude (or modulus) of a vector a = (x, y) is its length, calculated using Pythagoras' theorem:
|a| = √(x² + y²)
Example 1: |a| where a = (3, 4) |a| = √(9 + 16) = √25 = 5
Example 2: |v| where v = (-5, 12) |v| = √(25 + 144) = √169 = 13
Example 3: |w| where w = (1, 1) |w| = √(1 + 1) = √2
The direction of a vector (x, y) is the angle θ it makes with the positive x-axis. It can be found using:
tan θ = y/x
You must consider which quadrant the vector lies in to find the correct angle.
Example: Find the direction of a = (3, 3). tan θ = 3/3 = 1, so θ = 45°
Example: Find the direction of b = (-1, √3). tan θ = √3/(-1), so the reference angle is 60°. Since the vector is in the second quadrant (negative x, positive y), θ = 180° - 60° = 120°.
A unit vector has magnitude 1. To find the unit vector in the direction of a, divide a by its magnitude:
â = a / |a|
Example: Find the unit vector in the direction of a = (3, 4).
|a| = 5, so â = (3/5, 4/5) = (0.6, 0.8)
Check: |â| = √(0.36 + 0.64) = √1 = 1 ✓
Key Point: Unit vectors are useful for specifying direction without magnitude. To get a vector of any length r in the same direction as a, multiply the unit vector by r: râ.
To add two vectors, add their corresponding components:
a + b = (a₁ + b₁, a₂ + b₂)
Example: (2, 5) + (3, -1) = (5, 4)
Vector addition follows the triangle rule or the parallelogram rule:
To subtract vectors, subtract corresponding components:
a - b = (a₁ - b₁, a₂ - b₂)
Example: (5, 3) - (2, 7) = (3, -4)
a - b is the vector from the head of b to the head of a (when both start from the same point).
Multiplying a vector by a scalar k multiplies each component:
ka = k(a₁, a₂) = (ka₁, ka₂)
Example: 3 × (2, -1) = (6, -3)
| Value of k | Effect |
|---|---|
| k > 1 | Vector is stretched (longer), same direction |
| 0 < k < 1 | Vector is compressed (shorter), same direction |
| k = -1 | Vector is reversed (opposite direction, same magnitude) |
| k < 0 | Vector is reversed and scaled |
Two vectors are equal if they have the same magnitude and the same direction. In component form: a = b if and only if a₁ = b₁ and a₂ = b₂.
Two vectors are parallel if one is a scalar multiple of the other:
b = ka for some scalar k
Example: a = (2, 3) and b = (6, 9) are parallel because b = 3a.
Example: p = (4, -2) and q = (-6, 3) are parallel because q = -1.5p.
Exam Tip: To check if two vectors are parallel, check if the ratios of corresponding components are equal: a₁/b₁ = a₂/b₂. Alternatively, check if one is a scalar multiple of the other.
Two forces act on an object: F₁ = (5, 3) N and F₂ = (-2, 4) N. Find the resultant force and its magnitude.
Resultant = F₁ + F₂ = (5 + (-2), 3 + 4) = (3, 7) N
Magnitude = √(9 + 49) = √58 ≈ 7.62 N
Given A(2, 1) and B(5, 6), find: (a) the vector AB (b) the magnitude of AB (c) the unit vector in the direction of AB
(a) AB = (5 - 2, 6 - 1) = (3, 5)
(b) |AB| = √(9 + 25) = √34
(c) Unit vector = (3/√34, 5/√34)
A boat moves with velocity v = (4, 3) km/h. Find the speed of the boat and the direction of travel.
Speed = |v| = √(16 + 9) = √25 = 5 km/h
Direction: tan θ = 3/4, so θ = arctan(0.75) ≈ 36.9° from the positive x-direction.