Scalar Projection

Scalar Projection

The scalar projection of $\mathbf{a}$ onto $\mathbf{b}$ answers: "how much of $\mathbf{a}$ points in the direction of $\mathbf{b}$?"

$\text{comp}_{\mathbf{b}}(\mathbf{a}) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|}$

This is the signed length of the shadow of $\mathbf{a}$ onto the line through $\mathbf{b}$.

Derivation

From the dot product formula: $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| \cdot |\mathbf{b}| \cdot \cos(\theta)$

Dividing by $|\mathbf{b}|$:

$\text{comp}_{\mathbf{b}}(\mathbf{a}) = |\mathbf{a}| \cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|}$

Vector Projection

The vector projection (not required here) returns the actual vector:

$\text{proj}_{\mathbf{b}}(\mathbf{a}) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2} \cdot \mathbf{b}$

Examples

Your Task

Implement double scalar_proj(double ax, double ay, double az, double bx, double by, double bz) that returns the scalar projection of $\mathbf{a}$ onto $\mathbf{b}$.

Use #include <math.h> for sqrt.