Kuzee
Vector Algebra and Analytic Geometry of Space

The Historical Background of Vectors

A Simple Historical Scenario

The Problem: Describing Motion

Imagine you are a 17th-century physicist facing a deceptively simple question: how do you describe the motion of an object precisely?

By then people already knew that a moving object has:

  • a position (where it is)
  • a speed (how fast it moves)
  • a direction (which way it moves)

But how can one mathematical tool capture all of this at once?

The Problem with Traditional Methods

The ancient Greek mathematician Euclid described motion geometrically:

  1. Use a point for position
  2. Use a segment for distance
  3. Use an angle for direction

The limitations:

  • Position, distance, and direction could only be described separately
  • No single unified tool represented motion
  • Computations were cumbersome, especially for several moving objects

The Mathematicians’ Breakthrough

One day, mathematicians noticed something remarkable:

If you use an arrow to represent motion, then:

  1. The length of the arrow = the speed (magnitude)
  2. The direction of the arrow = the direction of motion
  3. The start of the arrow = the position of the object

With a single arrow you can capture all the information about a motion!

The Birth of Vectors

Mathematicians realized: a directed segment can represent any quantity that has both magnitude and direction!

This is the origin of the vector.

Exercises

Exercise 1

Use a vector to represent the displacement of an object from point A(1,2) to point B(4,6).

vectordisplacement

Approach: the displacement vector equals the endpoint coordinates minus the start coordinates.

Steps:

  1. AB⃗=(4−1, 6−2)=(3, 4)\vec{AB} = (4-1,\ 6-2) = (3,\ 4)
  2. Its magnitude is ∣AB⃗∣=32+42=5|\vec{AB}| = \sqrt{3^2 + 4^2} = 5
  3. Its direction: it points 4 units up for every 3 units right.

Answer: AB⃗=(3,4)\vec{AB} = (3, 4), with magnitude 55.

Exercise 2

A boat moves 30 km due east, then 40 km due north. Find its total displacement.

vectoradditionresultant

Approach: represent each leg as a vector and add them.

Steps:

  1. First leg: a⃗=(30,0)\vec{a} = (30, 0)
  2. Second leg: b⃗=(0,40)\vec{b} = (0, 40)
  3. Total displacement: a⃗+b⃗=(30,40)\vec{a} + \vec{b} = (30, 40)
  4. Magnitude: ∣a⃗+b⃗∣=302+402=50|\vec{a}+\vec{b}| = \sqrt{30^2 + 40^2} = 50 km

Answer: the boat is 50 km from the start, in the direction 36.87°36.87° north of east (since tan⁡θ=4030\tan\theta = \frac{40}{30}).

Next
The Basic Concepts of Vectors

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