03 · Types of Vectors

Overview of All Types

Vectors are classified based on their magnitude, direction, position, and relationship with other vectors. Here is every type you need to know:

★ High Priority

0

Zero Vector (Null Vector)

A vector whose magnitude is 0. Its direction is undefined / indeterminate. Denoted 0.

Example: AA = 0 (same start and end)

★ High Priority

â

Unit Vector

A vector with magnitude = 1. Used to indicate direction only. Written with a "hat": â, b̂.

î, ĵ, k̂ are standard unit vectors

↗↗

Co-initial Vectors

Two or more vectors that have the same initial point (tail). May differ in direction and magnitude.

OA and OB both start from O

∥

Collinear Vectors

Vectors that are parallel to the same line. They may point in the same or opposite directions.

a = 2î and b = −5î

★ Important

= =

Equal Vectors

Vectors with the same magnitude AND same direction. Position doesn't matter — they can start anywhere.

Any two arrows of same length & direction

★ Important

-a

Negative of a Vector

A vector with the same magnitude but opposite direction. If a goes A→B, then −a goes B→A.

BA = −AB

→→

Like & Unlike Vectors

Like vectors: same direction (parallel, not antiparallel).
Unlike vectors: opposite directions.

Like: both pointing East

🔄

Free Vector

A vector not fixed to any position in space. Only magnitude and direction matter, not starting point.

Most vectors in algebra are free vectors

Zero Vector — Deep Dive

The zero vector 0 is special — its magnitude is 0, but its direction is not defined. It plays the role of additive identity in vector algebra.

Properties of Zero Vector a + 0 = a (additive identity)
a + (−a) = 0 (additive inverse)
λ · 0 = 0 for any scalar λ
0 · a = 0 (multiply by zero scalar)

📌 Common Mistake

The zero vector 0 is different from the scalar 0. Do not confuse them. 0 has no direction; that's what makes it special.

Unit Vector — In Detail

The unit vector â along a gives the pure direction of a without its size.

Formula â = a / |a|, where |â| = 1

Standard unit vectors:
|î| = |ĵ| = |k̂| = 1
î·î = ĵ·ĵ = k̂·k̂ = 1 (self dot = 1)

✏️ Example

Find unit vector along p = 12î + 3ĵ − 4k̂

|p| = √(144 + 9 + 16) = √169 = 13

p̂ = (12/13)î + (3/13)ĵ + (−4/13)k̂

Equal vs Collinear vs Parallel

Type	Magnitude	Direction	Position
Equal Vectors	Same	Same	Anywhere
Collinear Vectors	Any	Same or Opposite	Parallel lines
Like Vectors	Any	Same	Parallel
Unlike Vectors	Any	Opposite	Antiparallel
Co-initial	Any	Any	Same start point

🎯 JEE Tip

Equal vectors do NOT need to start at the same point. If AB = CD, it just means ABDC forms a parallelogram. This geometric fact is used frequently in JEE problems.

Visualizing Negative Vector

✏️ Example

If a = AB= 3î + 2ĵ, then −a = AB= −3î − 2ĵ

Both have magnitude = √(9+4) = √13. Only direction is reversed.

Co-initial Vectors

Co-initial vectors originate from the same point. They may go in completely different directions.

Collinear Vectors — Formal Definition

Two vectors are collinear if and only if they are scalar multiples of each other.

Condition a = λb for some real λ

Component form: a₁/b₁ = a₂/b₂ = a₃/b₃

If λ > 0 → same direction (like vectors)
If λ < 0 → opposite direction (unlike vectors)

✏️ Example

Are a = 6î − 9ĵ + 12k̂ and b = 2î − 3ĵ + 4k̂ collinear?

Check: 6/2 = −9/−3 = 12/4 = 3 → Yes! a = 3b ✓

🎯 Board + JEE

To check if 3 points A, B, C are collinear: Find AB and AC. If AB = λ · AC, the points are collinear. This is a 2-mark question in boards and quick step in JEE multi-part problems.

Quick Summary — All Types

Type	Key Feature	Notation
Zero / Null	Magnitude = 0, direction undefined	0
Unit	Magnitude = 1	â, î, ĵ, k̂
Co-initial	Same starting point	OA, OB
Equal	Same magnitude and direction	a = b
Negative	Same magnitude, opposite direction	−a
Collinear	Parallel (same or opposite direction)	a = λb
Like	Same direction	λ > 0
Free	Position doesn't matter	Most algebra vectors