Vector Subtraction
Subtract vectors component by component · 벡터 뺄셈
1. What Is Vector Subtraction?
Vector subtraction finds the difference between two vectors. The cleanest way to think about it: subtracting b is the same as adding the negative of b.
a − b = a + (−b)
So every rule you know from vector addition carries over — you simply flip the sign of every component of b first. As with addition, both vectors must have the same dimension.
벡터 뺄셈은 두 벡터의 차이를 구합니다. b를 빼는 것은 −b를 더하는 것과 같습니다(a − b = a + (−b)). 따라서 b의 모든 성분 부호를 바꾼 뒤 더하면 되며, 두 벡터의 차원이 같아야 합니다.
2. The Component Rule
Subtract corresponding entries, one axis at a time:
a − b = (a₁ − b₁, a₂ − b₂, …, aₙ − bₙ)
Worked example (3D). For a = (5, 7, 9) and b = (1, 2, 3):
a − b = (5−1, 7−2, 9−3) = (4, 5, 6)
Worked example (2D). For a = (4, 3) and b = (1, 1): a − b = (3, 2).
대응하는 성분끼리 뺍니다: a − b = (a₁−b₁, …, aₙ−bₙ). 예: (5, 7, 9) − (1, 2, 3) = (4, 5, 6), (4, 3) − (1, 1) = (3, 2).
3. The Geometric Picture
Draw a and b from the same origin. The difference a − b is the arrow that points from the tip of b to the tip of a. A useful memory hook: the result points toward the first vector (the minuend).
This is exactly why displacement, relative velocity, and “B minus A” position differences all use subtraction.
a와 b를 같은 원점에서 그리면, a − b는 b의 끝점에서 a의 끝점으로 향하는 화살표입니다(앞 벡터 쪽을 가리킴). 변위, 상대속도, 위치 차이가 모두 이 뺄셈을 사용합니다.
4. Order Matters & Key Properties
- Not commutative. a − b ≠ b − a in general.
- Reversing flips the sign. b − a = −(a − b) — same length, opposite direction.
- A vector minus itself is zero. a − a = 0.
- Distributes with scalars. k(a − b) = ka − kb.
For the 3D example, b − a = (−4, −5, −6) = −(a − b), confirming the reversal rule.
뺄셈은 교환법칙이 성립하지 않습니다(a − b ≠ b − a). 순서를 바꾸면 부호가 뒤집힙니다: b − a = −(a − b). 또한 a − a = 0, k(a − b) = ka − kb. 위 예에서 b − a = (−4, −5, −6) = −(a − b).
5. Frequently Asked Questions
How do you subtract two vectors? Subtract their corresponding components: (a₁−b₁, a₂−b₂, …), or equivalently add the negative of the second vector.
Does the order matter in vector subtraction? Yes. a − b and b − a are negatives of each other — same magnitude, opposite direction.
What does a − b represent geometrically? The vector from the tip of b to the tip of a when both are drawn from the same starting point.
성분끼리 빼거나(또는 −b를 더해) 계산합니다. 순서가 중요하며 a − b와 b − a는 서로 부호가 반대입니다. 기하적으로 a − b는 b의 끝에서 a의 끝으로 향하는 벡터입니다.
Ready to practice? Drill vector subtraction and the rest of linear algebra on C:Vector, or review the full vector reference and the related vector addition and scalar multiplication.