Mathematics.

vector basics

Vectors

Linear Algebra30 minDifficulty3 out of 10

You should know: coordinate plane

Overview

A vector is a quantity with both magnitude and direction, represented as an arrow in space or, algebraically, as an ordered list of components. Vectors let us describe displacement, velocity, and force in a way that plain numbers (scalars) cannot — knowing a car moves at '60' isn't enough; you need '60 mph north'.

Intuition

A scalar is a single number: temperature, mass, distance traveled. A vector is that plus a direction: 5 km north, 3 newtons downward. Two vectors add by placing them head to tail — walk 3 blocks east then 4 blocks north, and your net displacement is a single diagonal vector of length 5.

Interactive Graph

Drag vector endpoints and see the sum/dot product update

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Formal Definition

Definition

In ℝⁿ, a vector is an ordered n-tuple, with addition and scalar multiplication defined componentwise:

v⃗=(v1,v2,…,vn)∈Rn\vec{v} = (v_1, v_2, \ldots, v_n) \in \mathbb{R}^n
Definition
u⃗+v⃗=(u1+v1,…,un+vn)\vec{u} + \vec{v} = (u_1+v_1, \ldots, u_n+v_n)

Vector addition

cv⃗=(cv1,…,cvn)c\vec{v} = (cv_1, \ldots, cv_n)

Scalar multiplication

∥v⃗∥=v12+⋯+vn2\|\vec{v}\| = \sqrt{v_1^2 + \cdots + v_n^2}

Magnitude (length) of a vector

Notation

NotationMeaning
v⃗\vec{v}A vector (also written in bold, v)
∥v⃗∥\|\vec{v}\|The magnitude (norm/length) of v
u⃗⋅v⃗\vec{u} \cdot \vec{v}The dot product of u and v

Properties

Dot product formula

u⃗⋅v⃗=u1v1+⋯+unvn=∥u⃗∥∥v⃗∥cos⁡θ\vec{u} \cdot \vec{v} = u_1v_1 + \cdots + u_nv_n = \|\vec{u}\|\|\vec{v}\|\cos\theta

Commutativity of addition

u⃗+v⃗=v⃗+u⃗\vec{u} + \vec{v} = \vec{v} + \vec{u}

Orthogonality test

u⃗⊥v⃗  ⟺  u⃗⋅v⃗=0\vec{u} \perp \vec{v} \iff \vec{u} \cdot \vec{v} = 0

Applications

Force, velocity, acceleration, and momentum are all vector quantities combined by vector addition.

Worked Examples

  1. Add componentwise.

    u⃗+v⃗=(2−1,3+5)=(1,8)\vec{u}+\vec{v} = (2-1, 3+5) = (1, 8)

Answer: (1, 8)

Practice Problems

Difficulty 3/10

Are u = (2, -3) and v = (6, 4) orthogonal?

Difficulty 5/10

Two cables pull on a bracket: F₁ = (300, 0) N (horizontal) and F₂ = (0, 400) N (vertical). Find the magnitude and direction of the resultant force the bracket must resist.

Difficulty 5/10

A 600 N force acts along a roof rafter inclined at 30° to the horizontal. Resolve it into horizontal and vertical components.

Difficulty 5/10

A tugboat pulls a barge with force F = (500, 200) N while the barge moves along displacement d = (10, 0) m. Compute the work done, W = F·d.

Common Mistakes

Common Mistake

Treating vector magnitude like simple addition of components.

‖(3,4)‖ ≠ 3+4. The magnitude uses the Pythagorean formula √(3²+4²) = 5, not the sum of components.

Quiz

Two forces (300, 0) N and (0, 400) N act on a joint. The magnitude of the resultant is:
The dot product F·d computes the work done by a force because it extracts:
Two vectors are orthogonal (perpendicular) exactly when:

Summary

  • A vector has both magnitude and direction, unlike a scalar.
  • In ℝⁿ, vectors add and scale componentwise.
  • Magnitude: ‖v‖ = √(v₁²+...+vₙ²); dot product: u·v = Σuᵢvᵢ = ‖u‖‖v‖cos θ.
  • Two vectors are orthogonal exactly when their dot product is zero.

References

  1. WebsiteWikipedia — Euclidean vector