Understanding Vector Projections

Vector projection calculates the shadow one vector casts onto another, fundamental in physics for work calculations, computer graphics for shading, and engineering for force decomposition.

Calculation Steps

Compute dot product: A·B = A₁B₁ + A₂B₂ + ... + A_nB_n
Calculate magnitude squared of B: ||B||² = B·B
Projection vector: proj_BA = (A·B / ||B||²) × B
Scalar projection: |proj_BA| = |A·B| / ||B||

Example Calculation

For A = [3, 4] and B = [1, 0]:

Dot product: 3×1 + 4×0 = 3
||B||² = 1² + 0² = 1
Projection: (3/1) × [1, 0] = [3, 0]
Magnitude: |3| = 3

Vector Operation	Formula
Dot Product	A·B = Σ(A_iB_i)
Projection	proj_BA = (A·B / B·B) × B
Angle Between	θ = cos⁻¹(A·B / (\|\|A\|\| \|\|B\|\|))

Frequently Asked Questions

What does a negative dot product indicate?

A negative dot product indicates vectors point in opposing directions (angle > 90°)

How are projections used in computer graphics?

Projections calculate lighting intensities by projecting surface normals onto light direction vectors

What's the difference between scalar and vector projection?

Scalar projection gives magnitude while vector projection provides both magnitude and direction

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