Vectors — Summary

1 What is a Vector?

A vector is a quantity that has both magnitude (size) and direction. It is represented geometrically as a directed line segment (arrow).

Scalar vs Vector Quantities

Scalar (magnitude only)	Vector (magnitude + direction)
Mass, time, speed, distance, temperature	Displacement, velocity, force, acceleration, weight
Example: 40 km/h is a speed	Example: 40 km/h East is a velocity
Example: 3 miles is a distance	Example: 3 miles North is a displacement

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Speed and velocity are NOT the same. Distance and displacement are NOT the same. The direction is what makes the difference.

2 Representing Vectors

Notation

A vector from A to B is written AB⃗ or as a bold letter v.
The magnitude (length) is written ‖v‖ or |AB|.
Component form: [x, y] — the horizontal and vertical components.

Direction on the Compass

Starting at East (0°) and going counter-clockwise:

Direction	Degrees
East	0°
North	90°
West	180°
South	270°

Example: "40° South of East" = start at East, go 40° toward South → orientation = 360° − 40° = 320°.
Example: "W 50° S" = start at West (180°), go 50° toward South → orientation = 180° + 50° = 230°.

Finding Components from Magnitude and Angle

x-component (horizontal)

x = ‖v‖ · cos(β) where β = angle of orientation

y-component (vertical)

y = ‖v‖ · sin(β)

Magnitude from components

‖v‖ = √(x² + y²)

3 Vector Operations

Addition and Subtraction

Add component-wise: [x₁, y₁] + [x₂, y₂] = [x₁+x₂, y₁+y₂].
Subtract component-wise: [x₁, y₁] − [x₂, y₂] = [x₁−x₂, y₁−y₂].
Graphically: head-to-tail method (place tail of second at head of first; resultant goes from tail of first to head of second).

Scalar Multiplication

k · [x, y] = [kx, ky] — stretches or shrinks the vector by k.
If k < 0, the direction reverses.

Special Vector Types

Type	Definition
Opposite vectors	Same magnitude, opposite direction: if v = [x, y] then −v = [−x, −y]
Equivalent vectors	Same magnitude and direction (position does not matter)
Orthogonal (perpendicular)	Dot product = 0
Collinear	One vector is a scalar multiple of the other; they point in the same or opposite direction

4 The Scalar (Dot) Product

The dot product takes two vectors and returns a number (scalar). It measures how much one vector points in the direction of another.

Using components

u · v = x₁x₂ + y₁y₂

Using magnitudes & angle

u · v = ‖u‖ · ‖v‖ · cos(θ)

Finding angle between vectors

cos(θ) = (u · v) / (‖u‖ · ‖v‖)

Key Properties

If u · v = 0, the vectors are orthogonal (perpendicular).
If u · v = ‖u‖ · ‖v‖, the vectors point in exactly the same direction (θ = 0°).
If u · v = −‖u‖ · ‖v‖, the vectors point in opposite directions (θ = 180°).
The dot product is commutative: u · v = v · u.

✏️

Example: u = [6, 8], v = [5, 12]
u · v = 6×5 + 8×12 = 30 + 96 = 126
‖u‖ = √(36+64) = 10, ‖v‖ = √(25+144) = 13
cos θ = 126/(10×13) = 0.969 → θ ≈ 14.5°

5 Applications

Vectors model any situation involving magnitude and direction: navigation, forces, velocity.

Step 1: Always start with a clear sketch labelling all known magnitudes and directions.
Step 2: Break each vector into [x, y] components.
Step 3: Add/subtract components to find resultant.
Step 4: Convert back to magnitude and direction using ‖v‖ = √(x²+y²) and β = arctan(y/x).

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Work example: The dot product W = F · d gives the work done when force F is applied over displacement d. Only the component of F in the direction of d does work.

6 Common Mistakes

Mistake	Fix
Confusing speed and velocity (scalar vs vector)	Speed is a scalar (magnitude only). Velocity includes direction.
Wrong angle orientation	The standard orientation starts at East = 0° and goes counter-clockwise. "40° S of E" is not 40° — it is measured FROM East.
Adding magnitudes instead of components	You CANNOT add ‖u‖ + ‖v‖ to get ‖u + v‖. Add components: [x₁+x₂, y₁+y₂].
Dot product gives a vector	No — u · v is a scalar (a number), not a vector.
Using degrees vs radians in cos(θ)	Make sure your calculator is in the correct mode.