Introduction to
Vectors
Understand quantities with both magnitude and direction — the language of physics and modern mathematics.
In the physical world, some quantities are fully described by a single number (their size), while others need both a size and a direction to make complete sense.
Described by magnitude only. No direction needed.
Mass · Temperature · Speed · Time · Distance · Energy · Work
Described by magnitude + direction. Both are essential.
Displacement · Velocity · Force · Acceleration · Momentum
Speed (scalar) tells how fast. Velocity (vector) tells how fast AND in which direction. This tiny difference creates entirely different mathematics!
A vector is drawn as a directed line segment — an arrow. The length gives the magnitude; the arrowhead gives the direction.
Magnitude: |AB| or |a| or just a (italic)
- The starting point A is called the Initial Point (Tail)
- The ending point B is called the Terminal Point (Head)
- Length of arrow = magnitude of vector
- Arrowhead direction = direction of vector
| Property | Scalar | Vector |
|---|---|---|
| Defined by | Magnitude only | Magnitude + Direction |
| Written as | Plain number (5 kg) | Bold / arrow (a⃗, F) |
| Addition | Simple algebra | Triangle / Parallelogram law |
| Negative value | Less than zero | Opposite direction |
| Examples | 5 m, 10 kg, 25°C | 5 m East, 10 N upward |
| Multiplication | Algebraic product | Scalar product or cross product |
The position vector of point P is the vector OP, drawn from the origin O to P. It completely describes P's location in space.
Magnitude: |OP| = √(x² + y² + z²)
In 2D: If P = (x, y) then: OP = x î + y ĵ, |OP| = √(x² + y²)
P = (3, 4, 0): OP = 3î + 4ĵ
|OP| = √(9 + 16) = √25 = 5 units
Unit vector along +x axis. |î| = 1
Unit vector along +y axis. |ĵ| = 1
Unit vector along +z axis. |k̂| = 1
- Newton's 2nd Law: F = ma — force and acceleration are both vectors
- Work: W = F · d (dot product — scalar result)
- Torque: τ = r × F (cross product — vector result)
- Electric Field: E is a vector field — direction matters everywhere
- 3D Geometry: Lines and planes are elegantly described using vectors
Vectors appear in 3D Geometry, Mechanics, Electromagnetism, and Optics in JEE. Mastering Chapter 10 directly unlocks 3D Geometry. The dot product and cross product come up every single year — treat them as high-priority topics.
| Term | Meaning |
|---|---|
| Scalar | Magnitude only (no direction) |
| Vector | Magnitude + Direction |
| Magnitude | Length/size of vector, always ≥ 0 |
| Unit Vector (â) | Vector with magnitude exactly = 1 |
| Zero Vector (0⃗) | Magnitude = 0, direction undefined |
| Position Vector | Vector from origin O to point P |
| Free Vector | Vector not tied to any specific position |
CBSE asks 1–2 mark questions on definitions and component writing. Practice writing vectors like 3î − 2ĵ + 5k̂ and computing their magnitudes quickly.