Vectors
2.1 The Geometric Approach
Definition of vectors, magnitude, direction, triangle rule for addition, Cartesian components, unit vectors i j k, position vectors, and basic vector operations.
2.1 The Geometric Approach
Definition 2.1 — Vector: A quantity with both magnitude (size) and direction. Represented graphically as an arrow. Examples: velocity, force, acceleration.
Definition 2.2 — Initial & Terminal points: Vector AB starts at point A (initial) and ends at B (terminal).
Definition 2.3 — Magnitude: Written |AB|, |a|, or simply a. The length of the arrow.
Definition 2.4 — Equal vectors: Two vectors are equal if they have the same magnitude and same direction — they may have different positions in space.
Vector Addition — Triangle Rule
Definition 2.5 — Triangle rule: Place vectors head-to-tail. The sum c = a + b connects the tail of a to the head of b.
Example 2.1 — Ship navigation
Ship speed in still water = 30 km/h north. Current = 20 km/h east.
(a) Pointing north
α = arctan(2030) = arctan(23)
33.7° east of north
(b) Travel due north
ψ = arcsin(2030) = arcsin(23)
41.8° west of north
(c) Land speed
v = 30² − 20² = 500
22.4 km/h north
Cartesian Components
Unit vectors: i (x-axis), j (y-axis), k (z-axis) — each has magnitude 1.
Any vectora = a₁i + a₂j + a₃k
Magnitude|a| = a₁² + a₂² + a₃²
Example — Cartesian operations
a = 3i + 2j − k, b = i − 4j + 2k
Addition
a + b = (3 + 1)i + (2 − 4)j + (−1 + 2)k
4i − 2j + k
Magnitude
|a| = 3² + 2² + (−1²) = 9 + 4 + 1 = 14
≈ 3.742
Unit vector
â = a|a| = 3i + 2j − k14
direction of a
Scalar Multiplication and Key Properties
- a + b = b + a (commutative)
- (a + b) + c = a + (b + c) (associative)
- k(a + b) = ka + kb (distributive)
- a − b = a + (−b) = a + (−1)b
Position vector: Vector from origin O to point P = (x,y,z) is OP = xi + yj + zk.
Vector from A=(a₁,a₂,a₃) to B=(b₁,b₂,b₃): AB = (b₁−a₁)i + (b₂−a₂)j + (b₃−a₃)k
Vector from A=(a₁,a₂,a₃) to B=(b₁,b₂,b₃): AB = (b₁−a₁)i + (b₂−a₂)j + (b₃−a₃)k