Vector Calculator

Vector Input

Vector a: i + j

Vector b: i + j

Vector Input

Vector a: i + j + k

Vector b: i + j + k

Result

Vector Calculator Introduction

The Vector Calculator is a professional online mathematical tool designed specifically for handling 2D and 3D vector operations. It provides eight core vector calculation functions (Vector Addition, Vector Subtraction, Dot Product (Inner Product), Cross Product (Outer Product), Vector Magnitude, Vector Angle, Unit Vector, Projection Vector) that meet the needs of students, teachers, and engineers in mathematics, physics, and engineering computations.

How to Use the Vector Calculator

1. Select the calculation dimension — supports 2D and 3D calculations
2. Enter vector values; you can click the example button to see how to input
3. Select the operation type — there are eight types of calculations available
4. View results — you can see the results and calculation steps for each operation.

Vector Calculation Formulas

1. Vector Addition

a + b = (a₁ + b₁, a₂ + b₂, a₃ + b₃)

2. Vector Subtraction

a - b = (a₁ - b₁, a₂ - b₂, a₃ - b₃)

3. Vector Dot Product (Inner Product)

a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a| |b| cosθ

4. Vector Cross Product (Outer Product)

a × b = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)

5. Vector Magnitude

|a| = √(a₁² + a₂² + a₃²)

6. Vector Angle

cosθ = a · b
|a| |b|

7. Unit Vector

â = a
|a|

8. Projection Vector

proj_b a = (a · b)
|b|² b