Shadows of vectors
Imagine shining a light straight down onto the line of vector v. The shadow of vector u on that line is its projection.
Scalar projection
The scalar projection of u onto v is the signed length of the shadow. It uses the dot product:
comp_v u = (u · v) / |v|
It’s positive if the angle between u and v is acute, negative if obtuse.
Vector projection
The vector projection is the shadow as a vector, pointing along v:
proj_v u = [(u · v) / |v|²] v
Example
u = [4, 3], v = [5, 0]
- u · v = 20; |v| = 5
- Scalar projection: 20/5 = 4
- Vector projection: (20/25)[5, 0] = [4, 0]
The shadow of [4, 3] on the x-axis is [4, 0], as expected.
A slanted example
u = [2, 6], v = [3, 4]
- u · v = 6 + 24 = 30; |v| = 5
- Scalar projection: 30/5 = 6
- Vector projection: (30/25)[3, 4] = 1.2[3, 4] = [3.6, 4.8]
The orthogonal component
Every vector u can be split into two parts relative to v:
- Parallel: proj_v u
- Perpendicular (the orthogonal component): u − proj_v u
From the slanted example: [2, 6] − [3.6, 4.8] = [−1.6, 1.2]
Check: [−1.6, 1.2] · [3, 4] = −4.8 + 4.8 = 0 ✓
Applications
Component of force: A 100 N force pulls a sled at 25° above the ground. The component of force along the ground: 100 cos 25° ≈ 90.6 N. That’s a scalar projection onto the horizontal.
Inclined planes: The component of gravity along a ramp is a projection.
Work: W = F · d = (scalar projection of F onto d) × |d|.
Distance from a point to a line: use the orthogonal component’s length.
Computer graphics: projections create shadows and 3D-to-2D views.
Projection onto i, j, k
The scalar projections of u onto i, j and k are simply its components.
Worked examples
Example 1: Scalar projection of [3, 4] onto [1, 0]? 3.
Example 2: Vector projection of [1, 2] onto [1, 1]? (3/2)[1, 1] = [1.5, 1.5].
Example 3: Scalar projection of [−2, 1] onto [1, 0]? −2 (obtuse angle).
Common mistakes
- Dividing by |v| instead of |v|² in the vector projection.
- Projecting the wrong vector onto the other (proj_v u ≠ proj_u v in general).
- Forgetting the sign of the scalar projection.
Try it yourself
- Scalar projection of [6, 8] onto [3, 4]?
- Vector projection of [5, 0] onto [1, 1]?
- Orthogonal component of [5, 0] relative to [1, 1]?
- Component of a 40 N force at 60° to the horizontal, along the horizontal?
Answers: 1) 10 2) [2.5, 2.5] 3) [2.5, −2.5] 4) 20 N
Quick recap
- Scalar projection: (u · v)/|v|; vector projection: [(u · v)/|v|²] v.
- The orthogonal component is u minus its projection.
- Projections find the component of force in a direction.