Vector Projection
Drop one vector's shadow onto another · 벡터 투영
1. What Is Vector Projection?
The projection of a onto b is the “shadow” that a casts along the direction of b — the part of a that points the same way as b. Imagine a light shining straight down onto the line through b: the shadow of a is the projection.
It answers “how much of a lies along b?” and underlies decomposition, least-squares fitting, and the physics of work (force along a direction of motion).
a를 b에 투영한다는 것은 a가 b 방향으로 드리우는 "그림자", 즉 a 중 b와 같은 방향 성분을 뜻합니다. "a가 b 방향으로 얼마나 놓여 있는가"에 답하며, 분해·최소제곱·일(work) 계산의 토대입니다.
2. The Formula & a Worked Example
The vector projection of a onto b is a vector pointing along b:
proj_b a = ((a · b) / (b · b)) b
The scalar projection (the signed length of that shadow) is (a · b) / |b|.
Worked example. Project a = (3, 4) onto b = (1, 0):
- a · b = (3)(1) + (4)(0) = 3, and b · b = 1
- proj_b a = (3 / 1) (1, 0) = (3, 0)
- scalar projection = (a · b) / |b| = 3 / 1 = 3
This matches intuition: projecting onto the x-axis keeps the x-component (3) and drops the y-component (4).
벡터 투영: proj_b a = ((a · b)/(b · b)) b, 스칼라 투영(부호 있는 그림자 길이): (a · b)/|b|. 예: a = (3, 4)를 b = (1, 0)에 투영 → (3, 0), 스칼라 투영 = 3. x축 투영은 x성분만 남기고 y성분을 버립니다.
3. The Geometric Picture
Drop a perpendicular from the tip of a straight down to the line through b. The foot of that perpendicular marks the projection; the leftover piece, a − proj_b a, is perpendicular to b.
That split — a = (component along b) + (component perpendicular to b) — is the orthogonal decomposition of a relative to b.
a의 끝점에서 b의 직선으로 수선을 내리면 그 발이 투영이고, 나머지 a − proj_b a 는 b에 수직입니다. 즉 a = (b 방향 성분) + (b에 수직 성분)으로 분해됩니다(직교분해).
4. Key Properties
- Parallel to b. proj_b a always lies along the line of b.
- Perpendicular remainder. (a − proj_b a) · b = 0 — the leftover is orthogonal to b.
- Sign follows the dot product. If a · b < 0, the projection points opposite to b (scalar projection is negative).
- Onto a unit vector. If b is a unit vector (|b| = 1), the formula simplifies to proj_b a = (a · b) b.
투영 벡터는 b와 평행하고, 나머지는 b에 수직입니다((a − proj_b a) · b = 0). a · b < 0 이면 투영이 b와 반대 방향(스칼라 투영 음수). b가 단위벡터면 proj_b a = (a · b) b 로 간단해집니다.
5. Frequently Asked Questions
How do you project one vector onto another? Use proj_b a = ((a · b) / (b · b)) b — scale b by the ratio of the dot product to b’s length squared.
What is the difference between scalar and vector projection? The scalar projection (a · b)/|b| is just the signed length; the vector projection multiplies that direction back in to give a full vector.
What does a negative projection mean? The component of a along b points opposite to b — it happens exactly when the dot product a · b is negative.
proj_b a = ((a · b)/(b · b)) b 로 구합니다. 스칼라 투영은 부호 있는 길이((a · b)/|b|), 벡터 투영은 그 방향을 곱해 벡터로 만든 것입니다. 투영이 음수면 a의 b 방향 성분이 b와 반대임을 뜻합니다(a · b < 0).
Ready to practice? Drill projection and the rest of linear algebra on C:Vector, or review the full vector reference and the related dot product, magnitude, and unit vector.