➡️ 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

$\begin{bmatrix}
1 \\
2 \\
3
\end{bmatrix}$

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 x [1, 3, 5] = [2, 6, 10]

Used for:

• Scaling data
• Adjusting weights

4.Dot Product (Very Important)

$[1,\,2] \cdot [3,\,4] = (1 \times 3) + (2 \times 4) = 11$

Used in:

• ML predictions
• Similarity calculation
• Projections

🗣 Hinglish Tip: Dot product = multiply + add

5.Vector Magnitude (Length)

Formula:

$|v| = \sqrt{x^2 + y^2 + \cdots}$

Example:

|[3, 4]| = 5

Used in:

• Distance calculation
• Normalization

6.Unit Vector

A vector with magnitude = 1.

Formula:

$\text{unit\_vector} = \frac{v}{|v|}$

Used in:

• Direction only (no size)
• Physics & graphics

Vector Comparison

• Same dimension required
• Order matters
• Length can differ

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