Vectors

What is a Vector?

A vector is a quantity that has both magnitude (size) and direction. This distinguishes vectors from scalars, which have magnitude only.

Examples of vectors: displacement, velocity, force, acceleration

Examples of scalars: distance, speed, mass, temperature, time

A vector from point A to point B is written as AB (with an arrow above, or in bold). You can also write it as a single bold letter: a or v.

Column Vector Notation

In 2D, vectors are written as column vectors: (x, y) where:

• x = horizontal component (positive = right, negative = left)
• y = vertical component (positive = up, negative = down)

Example: The vector (3, 4) means 3 right and 4 up.

The vector from A(1, 2) to B(4, 6) is AB = (4-1, 6-2) = (3, 4)

General rule: Vector AB = position of B minus position of A = OB - OA

Adding Vectors

To add vectors, add the corresponding components:

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

Example: (2, 3) + (-1, 5) = (2+(-1), 3+5) = (1, 8)

Geometrically: place the vectors head to tail. The resultant vector goes from the tail of the first to the head of the last.

Triangle law: if you travel along vector a then vector b, the total displacement is a + b.

Subtracting Vectors

Subtract the corresponding components:

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

Example: (4, 2) - (1, 5) = (4-1, 2-5) = (3, -3)

Geometrically: a - b = a + (-b), where -b is the vector b reversed.

Scalar Multiplication

Multiply each component by the scalar:

k × (x, y) = (kx, ky)

Example: 3 × (4, -2) = (12, -6)

Effects of scalar multiplication:

• k > 1: vector gets longer, same direction
• 0 < k < 1: vector gets shorter, same direction
• k = -1: vector reverses direction (same magnitude)
• k < 0: vector reverses direction and changes length

Magnitude (Length) of a Vector

The magnitude of vector (x, y) is its length, found using Pythagoras:

|v| = √(x² + y²)

Example: |(3, 4)| = √(9 + 16) = √25 = 5

Example: |(-5, 12)| = √(25 + 144) = √169 = 13

A unit vector has magnitude 1. To make a unit vector in the direction of v: divide each component by |v|.

Parallel Vectors

Two vectors are parallel if one is a scalar multiple of the other — they point in the same or opposite directions.

Example: (2, 4) and (1, 2) are parallel because (2, 4) = 2 × (1, 2)

Example: (2, 6) and (1, 3) are parallel — the ratio of components is 2:1 for both.

Example: (2, 6) and (4, 6) are NOT parallel — 2/4 ≠ 6/6.

To check: are the components in the same ratio? If a/c = b/d (same ratio), the vectors (a, b) and (c, d) are parallel.

Equal Vectors

Two vectors are equal if they have the same magnitude AND the same direction — regardless of where they are positioned.

The vector (3, 4) drawn starting from (0, 0) and the same vector drawn starting from (5, 2) are equal — position does not matter for equality.

Position Vectors

A position vector describes the location of a point relative to the origin O.

If point A is at coordinates (3, 5), its position vector is OA = (3, 5).

Vector between two points: AB = OB - OA = position of B - position of A

Example: A = (3, 1), B = (7, 5) AB = (7-3, 5-1) = (4, 4)

Vector Proof and Geometry

Vectors are used to prove geometric properties.

Example: Prove that the midpoint M of AB has position vector (OA + OB) / 2.

OM = OA + AM = OA + (1/2)AB = OA + (1/2)(OB - OA) = (1/2)(OA + OB)

This is a standard result: the midpoint position vector is the average of the two endpoint position vectors.

Common Mistakes

• Forgetting that vectors have direction — (3, 4) and (-3, -4) are not the same vector
• Adding magnitudes instead of components: |(3, 4)| + |(1, 2)| ≠ |(4, 6)|
• Confusing AB and BA — they are negatives of each other: BA = -AB
• Saying two vectors are parallel only if they point in the same direction — parallel includes opposite directions (one is a negative multiple of the other)
• Squaring incorrectly in the magnitude formula: |(-5, 12)| — (-5)² = 25, not -25

Tips and Tricks

• To find AB: always do B minus A (destination minus origin)
• Parallel vectors: check if the ratios of x-components and y-components are equal
• Magnitude: always square both components (including negatives), add, then square root
• Adding vectors geometrically (head to tail) is a useful visual check
• A vector multiplied by -1 reverses its direction — useful when working backwards along a path