Vector Projections

Definition

A vector projection is the decomposition of one vector $\mathbf{u}$ into a component that is parallel to another vector $\mathbf{v}$ . It represents the “shadow” or effective influence of $\mathbf{u}$ in the direction of $\mathbf{v}$ .

Why It Matters

In any system with multiple forces, we only care about the component that acts in our direction of interest. Projection is the tool that extracts “effective power” from “total power,” allowing us to calculate work and resolve conflicts efficiently.

Core Concepts

Vector Projection: The result is a vector in the direction of $\mathbf{v}$ $v$ : $\text{proj}_{\mathbf{v}} \mathbf{u} = \left( \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^2} \right) \mathbf{v}$ $proj_{v} u = (\frac{u \cdot v}{∣ v ∣ ^{2}}) v$ .
- How to read: “Projection of u onto v equals (u dot v over magnitude of v squared) times v.”
- Meaning: The component of $\mathbf{u}$ lying along $\mathbf{v}$ ; use for work ( $\mathbf{F} \cdot \mathbf{d}$ ) and force resolution.
Scalar Component: The signed length of the projection, $\text{comp}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|}$ $comp_{v} u = \frac{u \cdot v}{∣ v ∣}$ .
- How to read: “Scalar component of u on v equals u dot v over magnitude of v.”
- Meaning: Signed length of the shadow; positive if $\mathbf{u}$ and $\mathbf{v}$ point roughly the same way.
Relationship: The vector projection is the scalar component multiplied by the unit vector of $\mathbf{v}$ $v$ : $\text{proj}_{\mathbf{v}} \mathbf{u} = (\text{comp}_{\mathbf{v}} \mathbf{u}) \frac{\mathbf{v}}{|\mathbf{v}|}$ $proj_{v} u = (comp_{v} u) \frac{v}{∣ v ∣}$ .
- How to read: “Projection equals scalar component times v over magnitude of v.”
- Meaning: Scalar length times unit direction gives the full projection vector.

Definition

Why It Matters

Core Concepts

Connected Concepts

Connected notes