Understanding Vector Projections
In linear algebra, a vector projection essentially casts a "shadow" of one vector onto another. Imagine a light source shining perpendicular to Vector B; the shadow that Vector A casts onto Vector B is its vector projection.
This concept is widely used in physics (resolving forces into components), computer graphics, and engineering to determine how much of one vector acts in the direction of another. Our calculator provides both the Vector Projection (which gives a directional vector) and the Scalar Projection (which gives the length of that shadow).
Vector Projection Formula
The vector projection of A onto B creates a new vector that is parallel to B. It is calculated by scaling Vector B by a specific factor derived from the dot product.
- A · B: The Dot Product of vectors A and B.
- ||B||: The magnitude (length) of vector B.
- ||B||²: The magnitude of B squared (which is simply B · B).
Scalar Projection & Rejection
Scalar Projection: Also known as the component of A along B, this is a scalar number representing the magnitude of the projected vector.
Orthogonal Rejection: This is the portion of Vector A that is completely perpendicular to Vector B.