Mathematics.

vectors

Vectors in the Plane

Analytic Geometry25 minDifficulty3 out of 10

You should know: coordinate plane

Overview

A vector in the plane is a quantity with both magnitude and direction, represented as an ordered pair ⟨a, b⟩ or drawn as an arrow from one point to another. Unlike a point, which marks a location, a vector represents displacement, velocity, or force — it can be translated anywhere in the plane without changing its identity, since only its length and direction matter. Vectors support addition, scalar multiplication, and the dot product, and every vector can be decomposed into components along the standard basis vectors i = ⟨1,0⟩ and j = ⟨0,1⟩.

Intuition

Picture a vector as an arrow: sliding it anywhere in the plane without rotating or stretching it doesn't change which vector it is, because a vector only encodes 'how far and which way,' not 'from where.' Adding two vectors tip-to-tail (placing the second vector's tail at the first vector's tip) traces out the same displacement as adding their components separately — this is the parallelogram rule made algebraic. The dot product measures how much two vectors point in the same direction: it's largest when they're parallel, zero when they're perpendicular, and negative when they point opposite ways.

Formal Definition

Definition

For vectors u = ⟨u₁, u₂⟩ and v = ⟨v₁, v₂⟩ and scalar k:

u+v=⟨u1+v1, u2+v2⟩\mathbf{u} + \mathbf{v} = \langle u_1+v_1,\ u_2+v_2 \rangle
Vector addition
ku=⟨ku1, ku2⟩k\mathbf{u} = \langle ku_1,\ ku_2 \rangle
Scalar multiplication
∥u∥=u12+u22\|\mathbf{u}\| = \sqrt{u_1^2+u_2^2}
Magnitude
u⋅v=u1v1+u2v2=∥u∥∥v∥cos⁡θ\mathbf{u}\cdot\mathbf{v} = u_1v_1 + u_2v_2 = \|\mathbf{u}\|\|\mathbf{v}\|\cos\theta
Dot product

Worked Examples

  1. Add corresponding components.

    u+v=⟨3+(−1), 4+2⟩=⟨2,6⟩\mathbf{u}+\mathbf{v} = \langle 3+(-1),\ 4+2\rangle = \langle 2, 6\rangle

  2. Apply the magnitude formula to u.

    ∥u∥=32+42=25=5\|\mathbf{u}\| = \sqrt{3^2+4^2} = \sqrt{25} = 5

Answer: u + v = ⟨2, 6⟩, ‖u‖ = 5

Practice Problems

Difficulty 3/10

Given u = ⟨2, −5⟩ and v = ⟨4, 1⟩, find 2u − v.

Difficulty 4/10

Determine whether u = ⟨6, −2⟩ and v = ⟨1, 3⟩ are perpendicular.

Difficulty 6/10

Two forces act on an object: F₁ = ⟨5, 0⟩ N and F₂ = ⟨0, 12⟩ N. Find the magnitude of the resultant (net) force.

Quiz

The dot product of two perpendicular vectors is:
For u = ⟨3,4⟩ and v = ⟨−1,2⟩, u + v equals:
A vector differs from a point because:

Summary

  • A vector ⟨a,b⟩ encodes magnitude and direction, not location; it can be freely translated in the plane.
  • Vector addition and scalar multiplication work componentwise; magnitude is √(a²+b²) via the Pythagorean theorem.
  • The dot product u·v = ‖u‖‖v‖cosθ measures alignment: positive when vectors point similarly, zero when perpendicular, negative when opposed.

References

  1. WebsiteWikipedia — Euclidean vector