Vector Operations with î and ĵ

Theory Explanation

This simulation demonstrates vector operations using unit vectors î (x-axis) and ĵ (y-axis):

1. Vector Magnitude and Direction

For any vector v = aî + bĵ:

• Magnitude: |v| = √(a² + b²)
• Direction: θ = tan⁻¹(b/a) with respect to x-axis

2. Dot Product

For vectors A = a₁î + a₂ĵ and B = b₁î + b₂ĵ:

• A·B = a₁b₁ + a₂b₂ = |A||B|cosθ
• When A·B = 0, vectors are perpendicular

3. Vector Projections

The component of A along B is given by:

(A·B)/|B|