Vectors

The fundamental building block of linear algebra

What is a Vector?

A vector is one of the most fundamental objects in mathematics. At its core, a vector represents a quantity that has both magnitude (how much) and direction (which way).

Think of it as an arrow pointing from one place to another. The arrow has a length (magnitude) and points in a specific direction. In two dimensions, we describe this arrow using two numbers: how far it goes horizontally (x) and how far it goes vertically (y).

Interactive: Drag the vector

Drag the tip of the vector to move it

Mathematically, we write a 2D vector as a column of numbers:

v⃗=[xy]\vec{v} = \begin{bmatrix} x \\ y \end{bmatrix}

The key insight is that a vector describes movement or displacement in space, not a fixed position. The vector [32]\begin{bmatrix} 3 \\ 2 \end{bmatrix} means move 3 units right and 2 units up from wherever you start.

Vector Addition

When we add two vectors, we are combining their movements. Imagine walking along vector a⃗\vec{a}, then from that endpoint, walking along vector b⃗\vec{b}. Where you end up is the sum a⃗+b⃗\vec{a} + \vec{b}.

This is called the tip-to-tail method: place the tail of the second vector at the tip of the first.

Vector Addition: Tip-to-Tail

a⃗=(2,1)\vec{a} = (2, 1)b⃗=(1,2)\vec{b} = (1, 2)a⃗+b⃗=(3,3)\vec{a} + \vec{b} = (3, 3)

Algebraically, we simply add the corresponding components:

[a1a2]+[b1b2]=[a1+b1a2+b2]\begin{bmatrix} a_1 \\ a_2 \end{bmatrix} + \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} = \begin{bmatrix} a_1 + b_1 \\ a_2 + b_2 \end{bmatrix}

Scalar Multiplication

A scalar is just a regular number. When we multiply a vector by a scalar, we scale the vector: stretching it if the scalar is greater than 1, shrinking it if between 0 and 1, and flipping its direction if negative.

Scalar Multiplication: Stretch, Shrink, Flip

1.0

Original vector

c⋅[xy]=[c⋅xc⋅y]c \cdot \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} c \cdot x \\ c \cdot y \end{bmatrix}

Notice how multiplying by 2 doubles the length, multiplying by 0.5 halves it, and multiplying by -1 reverses the direction completely. This simple operation underlies everything from computer graphics to physics simulations.

Vectors in Three Dimensions

Everything we have learned extends naturally to three dimensions. A 3D vector has three components: x, y, and z. The same operations of addition and scalar multiplication work the same way, just with an extra coordinate.

3D Vector: Drag to rotate the view

2.0
1.0
1.5

v⃗=(2.0,1.0,1.5)\vec{v} = (2.0, 1.0, 1.5)

v⃗=[xyz]\vec{v} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}

The 3D coordinate system uses three perpendicular axes. By convention, x points right, y points up, and z points toward you. This arrangement is called a right-handed coordinate system, and understanding it will become essential as we explore cross products and rotations in later chapters.

Looking Ahead

Vectors are the foundation. In the next chapter, we will combine them using linear combinations to understand what regions of space a set of vectors can reach. Later, we will package vectors into matrices and discover how they encode transformations of space itself.

Key Takeaways

  • A vector represents magnitude and direction, visualized as an arrow
  • Vectors are written as columns of numbers representing displacement along each axis
  • Vector addition combines movements using the tip-to-tail method
  • Scalar multiplication stretches, shrinks, or flips a vector
  • These concepts extend naturally from 2D to 3D and beyond