Vector Representation

PositionX-ComponentY-Component

📐What is a Vector?

A vector is a mathematical object that has both magnitude (length) and direction. Unlike a scalar which only has magnitude, vectors represent quantities like velocity, force, and displacement.

In the visualization above, you can drag the yellow point to see how the vector changes. The vector is represented by the yellow arrow from the origin (0,0) to your point.

🧩Breaking Down Components

X-Component (Horizontal)

The red arrow shows the horizontal component of the vector. This represents how far the vector extends along the X-axis. It's calculated as x = r × cos(θ), where r is the magnitude and θ is the angle.

Y-Component (Vertical)

The green arrow shows the vertical component of the vector. This represents how far the vector extends along the Y-axis. It's calculated as y = r × sin(θ), where r is the magnitude and θ is the angle.

📏Magnitude and Direction

Magnitude (Length)

The magnitude of a vector is its length, calculated using the Pythagorean theorem:

|v| = √(x² + y²)

Direction (Angle)

The angle θ (theta) is measured counterclockwise from the positive X-axis. It's calculated using the arctangent function:

θ = arctan(y / x)

The white arc in the visualization shows this angle from the X-axis

🎯 Vector Notation

Vectors can be written in multiple ways:

• (x, y) - Component form
• xi + yj - Unit vector notation
• r∠θ - Polar form

🔄 Conversion

You can convert between component form and polar form:

Cartesian → Polar:
r = √(x² + y²), θ = arctan(y/x)

Polar → Cartesian:
x = r×cos(θ), y = r×sin(θ)