Vectors in 2D

Definition

In the context of mathematics and physics, a Vector is a quantity that possesses both magnitude and direction. Geometrically, it is represented by a directed line segment; algebraically, it is represented as $\mathbf{v} = \langle a, b \rangle$ or $\mathbf{v} = a\mathbf{i} + b\mathbf{j}$, where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors in the $x$ and $y$ directions respectively.

• How to read: “The vector v equals the vector with components a and b, or a times i plus b times j.”
• Meaning: Two equivalent notations for the same 2D vector; components $(a, b)$ or basis expansion along $\mathbf{i}$ and $\mathbf{j}$.

Why It Matters

2D vectors are the gateway to spatial intelligence. They provide the bridge between pure numbers and physical direction. Without them, we cannot coordinate movement, align teams, or describe any action that happens in a plane.

Core Concepts

• Magnitude: The “length” of the vector, denoted by $\|\mathbf{v}\| = \sqrt{a^2 + b^2}$.

• How to read: “Norm of v equals square root of a-squared plus b-squared.”
• Meaning: Pythagorean length; always non-negative.

• Direction Angle: The angle $\alpha$ that the vector makes with the positive $x$-axis, where $\mathbf{v} = \|\mathbf{v}\| (\cos \alpha \mathbf{i} + \sin \alpha \mathbf{j})$.

• How to read: “v equals norm of v times (cosine alpha i plus sine alpha j).”
• Meaning: Polar form: magnitude times unit direction $(\cos\alpha, \sin\alpha)$.

• Unit Vector: A vector with a magnitude of 1. It represents pure direction. $\mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|}$.

• How to read: “u equals v divided by norm of v.”
• Meaning: Normalize by dividing out the length; preserves direction only.

• Operations:
  • Addition: $\langle a_1, b_1 \rangle + \langle a_2, b_2 \rangle = \langle a_1 + a_2, b_1 + b_2 \rangle$. Geometrically, this is the “Tip-to-Tail” rule.
  • How to read: “The sum of the vectors with components a one, b one, and a two, b two equals the vector with components a one plus a two, and b one plus b two.”
  • Meaning: Component-wise sum; tip-to-tail geometrically.
  • Scalar Multiplication: $k\langle a, b \rangle = \langle ka, kb \rangle$. This scales the magnitude without changing the direction (unless $k$ is negative).
  • How to read: “The scalar k times the vector a b equals the vector with components k a and k b.”
  • Meaning: Stretch by $|k|$; reverse direction if $k < 0$.

Connected Concepts

• Vectors in Space
• Angle Between Vectors
• Systems of Inequalities in 2D
• Unit Vectors
• First Principles Thinking