Concept units

Vectors: arrows made of numbers

Basic

Before you start

You need nothing else first. This unit starts from scratch.

The idea in one paragraph

A point on a map is a list of numbers: how far east and how far north it lies. A vector is the same kind of list, read as an arrow: "go 3 east and 1 north." You add two vectors by putting one arrow after the other, which means adding the lists number by number. You stretch a vector by multiplying every number by the same amount. Computers store points, directions and many other things this way, in 2D, in 3D or with many more numbers.

A point is a list of numbers

Pick a starting corner on a sheet of squared paper and call it the origin. Any other spot on the sheet is then two numbers: how many squares to the right, and how many squares up. These numbers are the spot's coordinates, written in brackets as (3, 1). The first number is called x and the second y.

A room needs a third number for height, called z. So a point in 3D is a list of three numbers, such as (1.5, −2, 4). Nothing changes in the rules below when the list gets longer.

A vector is an arrow that says how far to move

A vector is a list of numbers read as a move: (3, 1) means "3 to the right and 1 up." You can draw it as an arrow. The arrow's position on the page does not matter; only its length and direction do. Each number in the list is a component of the vector.

A point and a vector look the same on paper. A point (3, 1) is where you end up if you start at the origin and follow the arrow (3, 1).

Adding two vectors puts one arrow after the other
a b a + b
a = (3, 1)   b = (1, 2)   a + b = (4, 3)
Try this. Drag the round handles at the tips of a and b. Arrow b always starts where a ends. Goal: make a + b land on (5, 5) while a stays at (3, 1). Predict b first, then drag to check.

Adding vectors means walking one arrow and then the other. Put the tail of b at the tip of a. The arrow from the start of a to the tip of b is the sum a + b. In numbers, you add the matching components: the x's together, the y's together.

a + b = (a1 + b1, a2 + b2)

Here a1 is the first component of a and a2 its second. A bold letter stands for a whole vector. An italic letter with a small number stands for one of its components.

Multiplying by a number stretches the arrow

Multiply every component by the same number and the arrow keeps its direction but changes length. The number is called a scalar, because it scales the arrow. Doubling (3, 1) gives (6, 2), an arrow twice as long. A negative scalar also flips the arrow: −1 times (3, 1) is (−3, −1).

Subtracting is adding the flipped arrow. The arrow that leads from point a to point b is b − a, because a + (b − a) = b.

Worked example

Take a = (3, 1) and b = (1, 2).

  1. Add: (3 + 1, 1 + 2) = (4, 3).
  2. Stretch: 2 × (3, 1) = (2 × 3, 2 × 1) = (6, 2).
  3. The arrow from a to b: (1 − 3, 2 − 1) = (−2, 1). Check it: (3, 1) + (−2, 1) = (1, 2), which is b.

Your turn. With a = (2, 5) and b = (−1, 3), find a + b, 3a, and the arrow from a to b.

Answer

(1, 8); (6, 15); (−1 − 2, 3 − 5) = (−3, −2).

This is a simplification

This page treats a point and a vector as the same list of numbers, as most code does. Mathematicians keep them apart: adding two vectors makes sense, but adding two points (two places) does not. Code will happily add two points, so the meaning is up to you.

vectors.py: NumPy stores a vector as an array, and does the arithmetic component by component

import numpy as np

a = np.array([3, 1])
b = np.array([1, 2])
print(a + b)      # add: tip to tail
print(2 * a)      # stretch
print(b - a)      # the arrow from point a to point b

p = np.array([1.5, -2.0, 4.0])  # a point in 3D
print(p.shape)

[4 3]
[6 2]
[-2  1]
(3,)

Check yourself

  1. What is (2, −1) + (−3, 4)? Sketch both arrows tip to tail and check that the sum points where you expect.

    Answer

    (2 − 3, −1 + 4) = (−1, 3): one square left and three up.

  2. A drone starts at (1, 1, 0) and ends at (4, 5, 2). Which vector describes its trip?

    Answer

    End minus start: (4 − 1, 5 − 1, 2 − 0) = (3, 4, 2).

  3. Which scalar turns (4, −2) into (−2, 1)? What does it do to the arrow?

    Answer

    −0.5. It halves the length and flips the arrow to point the opposite way.

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