➡️ Vectors

Last Updated: Jan 2026

A vector is an ordered collection of numbers used to represent:

  • Data
  • Direction
  • Features
  • State

In programming, vectors are everywhere:

  • Arrays
  • Lists
  • Feature inputs
  • Coordinates

🗣 Hinglish Tip: Vector ko simple words me number ka ordered group samjho

Vector Representation

Mathematical Form

v = [2, 4, 6]

Programming Form

v = [2, 4, 6]

Each value is called a component.

Vector Dimension

The number of components in a vector is its dimension.

Examples:

  • [5] → 1D vector
  • [2, 3] → 2D vector
  • [1, 2, 3] → 3D vector

🗣 Hinglish Tip: Dimension = vector me kitne numbers hain

Vector Types

Row Vector

[ 1  2  3 ]

Used commonly in data rows.

Column Vector

[ 1 ]
[ 2 ]
[ 3 ]

Used in matrix multiplication.

Vector Operations

1.Vector Addition

Add corresponding elements.

[1, 2] + [3, 4] = [4, 6]

🗣 Hinglish Tip: Same index wale numbers add hote hain

2.Vector Subtraction

[5, 4] - [2, 1] = [3, 3]

3.Scalar Multiplication

Multiply vector by a number.

2 × [1, 3, 5] = [2, 6, 10]

Used for:

  • Scaling data
  • Adjusting weights

4.Dot Product (Very Important)

[1, 2] · [3, 4] = (1×3) + (2×4) = 11

Used in:

  • ML predictions
  • Similarity calculation
  • Projections

🗣 Hinglish Tip: Dot product = multiply + add

5.Vector Magnitude (Length)

Formula:

|v| = √(x² + y² + ...)

Example:

|[3, 4]| = 5

Used in:

  • Distance calculation
  • Normalization

6.Unit Vector

A vector with magnitude = 1.

Formula:

unit_vector = v / |v|

Used in:

  • Direction only (no size)
  • Physics & graphics

Vector Comparison

  • Same dimension required
  • Order matters
  • Length can differ
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