Basic Concepts of Vectors

I. Three Ways of Looking at a Vector

A vector (Vector) is understood differently in different disciplines. One of the keys to learning linear algebra is being able to switch freely between these three views.

1. The Physics View: a Vector Is an “Arrow”

In a physics class, a vector is an arrow in space:

• It has a direction and a length (magnitude)
• Examples: displacement, velocity, force

The essence of an arrow is determined by its direction and its length, not by where it happens to be drawn — after translating it, it is still the same vector.

2. The Computer Science View: a Vector Is an “Ordered Array”

To a programmer, a vector is an ordered list of numbers:

v = (3, 2)      ← a 2D vector (2 numbers)
w = (1, 0, 5)   ← a 3D vector (3 numbers)

Each number is a component of the vector on some coordinate axis.

3. The Mathematical View: a Vector Is an “Object You Can Operate On”

A mathematician abstracts a vector as an object satisfying certain rules of operation: it can be added, and it can be multiplied by a scalar (multiplied by a number). Whether it really is an “arrow” does not matter — what matters is how it is operated on.

II. Coordinates: Giving a Vector an “Address”

On the two-dimensional plane we set up a coordinate system (two perpendicular axes):

• The horizontal axis (the x-axis): the horizontal direction
• The vertical axis (the y-axis): the vertical direction

A vector can be written as a pair of coordinates:

v = (vx, vy)

Meaning: start at the origin, move vx units in the x direction, then move vy units in the y direction, and you arrive at the tip of the arrow.

• The vector (2, 3): 2 units to the right, 3 units up
• The start of a vector is always the origin (a convention of the geometric view)

III. Why an “Ordered Array”

The coordinates (2, 3) and (3, 2) are different vectors — the order matters. That is why a vector is called an “ordered array”:

(2, 3) ≠ (3, 2)    ← the first component lies on the x-axis, the second on the y-axis

In a higher-dimensional (n-dimensional) space, a vector is simply n ordered numbers:

v = (v₁, v₂, ..., vₙ)   ← an n-dimensional vector

IV. The Length (Norm) of a Vector

The length of a vector arrow is called its norm, written with two vertical bars:

$$\|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2} \quad \text{(2D)}$$

This is a direct application of the Pythagorean theorem — the arrow is the hypotenuse of a right triangle.

Example: v = (3, 4)
‖v‖ = √(3² + 4²) = √25 = 5

V. The Zero Vector and the Unit Vector

Common notation:

Zero vector: 0 = (0, 0, ..., 0)
Normalisation: v̂ = v / ‖v‖   (scale v to length 1, keeping its direction)

VI. Vector Addition: The Parallelogram Rule

Geometric Intuition

To add two vectors, use the parallelogram rule (or head-to-tail):

Move the start of v to the end of w (head to tail),
or draw v and w from the origin at the same time,
complete the parallelogram whose adjacent sides are the two of them,
and the diagonal arrow is v + w.

The Algebraic Rule

Coordinates are added component by component:

$\mathbf{v} + \mathbf{w} = (v_x + w_x,\; v_y + w_y)$

Example: v = (2, 1), w = (1, 3)
v + w = (2+1, 1+3) = (3, 4)

VII. Interactive Demo: Drag the Vectors and Watch Them Add

What to watch:

• Drag the sliders to change the components of v and w
• The orange arrow v+w always satisfies “add the coordinates component by component”
• The geometric position of v+w is the diagonal of the parallelogram

VIII. Scalar Multiplication: Stretching and Shrinking

Geometric Intuition

Scalar multiplication scales a vector by a scalar (a number):

2·v: stretch v to twice its length (direction unchanged)
0.5·v: shrink v to half its length (direction unchanged)
-1·v: reverse v (direction flipped, length unchanged)
-2·v: reverse it and stretch it to twice its length

        v  →  2v: longer
        v  →  0.5v: shorter
        v  →  -v: opposite direction

The Algebraic Rule

Every component is multiplied by that number:

$\lambda \mathbf{v} = (\lambda v_x,\; \lambda v_y)$

Example: v = (2, 1)
3v = (6, 3)
-2v = (-4, -2)

IX. What the Two Basic Operations Mean

Taken together, vector addition and scalar multiplication are the two basic operations of a linear combination. They satisfy a series of properties, and it is precisely those properties that define what “linear” means:

X. The Geometric Intuition of the Operations, Summarised

Summary

Symbols Used in This Article