Vector Projection

Vector Projection

The projection of v onto u is the component of v that lies in the direction of u:

$\text{proj}_{\mathbf{u}}\mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \, \mathbf{u}$

The scalar coefficient $\frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}}$ scales u to exactly the "shadow" of v along u.

Geometric Intuition

Imagine shining a light perpendicular to u — the shadow of v cast onto the line through u is exactly the projection. The perpendicular component $\mathbf{v} - \text{proj}_{\mathbf{u}}\mathbf{v}$ is orthogonal to u.

Examples

Note: For [1,2,3] onto [1,1,1]: the sum is 6, so the coefficient is $6/3 = 2$, giving [2, 2, 2].

Your Task

Implement dot(a, b) returning the dot product of two vectors, and project(v, u) returning the projection of v onto u.