Vector Operations Formulas
Essential vector formulas for dot product, cross product, and magnitude.
Learn vector math with clear explanations and worked examples.
Magnitude (Length) of a Vector
The magnitude of a vector tells you its length. For a 2D vector (v₁, v₂), the formula simplifies to |v| = √(v₁² + v₂²).
Dot Product (Scalar Product)
The dot product takes two vectors and returns a single number (a scalar). It measures how much two vectors point in the same direction.
When the dot product is zero, the vectors are perpendicular. When it is positive, they point in similar directions. When negative, they point in roughly opposite directions.
Cross Product (Vector Product)
The cross product takes two vectors and returns a new vector that is perpendicular to both. Its magnitude equals |a||b| sin θ, which is the area of the parallelogram formed by the two vectors.
The cross product only exists in 3D. The direction follows the right-hand rule: point your fingers from a toward b, and your thumb points in the direction of a × b.
Variables
| Symbol | Meaning |
|---|---|
| a, b | Vectors with components (a₁, a₂, a₃) and (b₁, b₂, b₃) |
| |v| | Magnitude (length) of vector v |
| θ | Angle between the two vectors |
| a · b | Dot product — a scalar (single number) |
| a × b | Cross product — a vector perpendicular to both a and b |
Example 1: Dot Product
Find the dot product of a = (3, −2, 5) and b = (1, 4, −3). Then find the angle between them.
Dot product: a · b = (3)(1) + (−2)(4) + (5)(−3) = 3 − 8 − 15 = −20
Magnitudes: |a| = √(9 + 4 + 25) = √38 ≈ 6.16
|b| = √(1 + 16 + 9) = √26 ≈ 5.10
cos θ = (a · b) / (|a||b|) = −20 / (6.16 × 5.10) = −20 / 31.42 = −0.6366
θ = arccos(−0.6366) ≈ 129.5° (the vectors point in roughly opposite directions)
Example 2: Cross Product
Find the cross product of a = (2, 3, 4) and b = (5, 6, 7).
First component: a₂b₃ − a₃b₂ = (3)(7) − (4)(6) = 21 − 24 = −3
Second component: a₃b₁ − a₁b₃ = (4)(5) − (2)(7) = 20 − 14 = 6
Third component: a₁b₂ − a₂b₁ = (2)(6) − (3)(5) = 12 − 15 = −3
a × b = (−3, 6, −3)
When to Use These
Vector operations appear throughout science, engineering, and computer graphics.
- Dot product: calculating work done by a force, projecting one vector onto another, checking if vectors are perpendicular
- Cross product: finding normal vectors to surfaces, calculating torque, determining the area of triangles in 3D
- Magnitude: finding distances, normalizing vectors to unit length, calculating speed from velocity components