Vector Addition
Add vectors component by component · 벡터 덧셈
1. What Is Vector Addition?
Vector addition combines two vectors into a single resultant vector. Algebraically it is the simplest operation in linear algebra: you just add the matching components. Geometrically it answers the question “if I travel along the first arrow, then along the second, where do I end up?”
Both vectors must have the same number of components — you can add two 3D vectors, but not a 2D vector and a 3D one. The result lives in the same space as the inputs.
벡터 덧셈은 두 벡터를 하나의 합 벡터로 합칩니다. 같은 위치의 성분끼리 더하면 되며, 두 벡터의 차원(성분 개수)이 같아야 합니다. 기하적으로는 "첫 화살표를 따라간 뒤 두 번째 화살표를 따라가면 어디에 도착하는가"에 대한 답입니다.
2. The Component Rule
Add corresponding entries, one axis at a time:
a + b = (a₁ + b₁, a₂ + b₂, …, aₙ + bₙ)
Worked example (3D). For a = (1, 2, 3) and b = (4, 5, 6):
a + b = (1+4, 2+5, 3+6) = (5, 7, 9)
Worked example (2D). For a = (1, 2) and b = (3, 1): a + b = (4, 3).
대응하는 성분끼리 더합니다: a + b = (a₁+b₁, …, aₙ+bₙ). 예: (1, 2, 3) + (4, 5, 6) = (5, 7, 9), (1, 2) + (3, 1) = (4, 3).
3. Tip-to-Tail & the Parallelogram Rule
To add geometrically, place the tail of b at the tip of a; the sum runs from the tail of a to the tip of b. Using the 2D example above, a = (1, 2) followed by b = (3, 1) lands at (4, 3):
Because order does not change the destination, drawing both vectors from a common origin forms a parallelogram whose diagonal is the sum — the same resultant, seen two ways.
기하적으로는 a의 끝점에 b의 시작점을 붙이면, 합은 a의 시작점에서 b의 끝점까지의 벡터입니다(삼각형법). 두 벡터를 같은 원점에서 그리면 평행사변형의 대각선이 합이 됩니다(평행사변형법). 위 예에서 (1, 2) + (3, 1) = (4, 3).
4. Key Properties
- Commutative. a + b = b + a — order does not matter.
- Associative. (a + b) + c = a + (b + c).
- Zero vector identity. a + 0 = a, where 0 = (0, …, 0).
- Additive inverse. a + (−a) = 0; negating flips every component.
- Distributes with scalars. k(a + b) = ka + kb.
These rules are exactly what make vectors a vector space — the foundation for everything from physics (adding forces) to graphics (combining displacements).
교환법칙 a + b = b + a, 결합법칙 (a + b) + c = a + (b + c), 영벡터 항등원 a + 0 = a, 덧셈 역원 a + (−a) = 0, 스칼라 분배 k(a + b) = ka + kb 가 성립합니다. 이 성질들이 벡터공간의 토대입니다.
5. Frequently Asked Questions
How do you add two vectors? Add their corresponding components: (a₁+b₁, a₂+b₂, …). The vectors must have the same dimension.
Can you add vectors of different dimensions? No. A 2D and a 3D vector have no component-by-component correspondence, so the sum is undefined.
Is vector addition commutative? Yes — a + b = b + a, which is why the tip-to-tail and parallelogram constructions give the same resultant.
두 벡터는 성분끼리 더하며 차원이 같아야 합니다. 차원이 다르면 덧셈은 정의되지 않습니다. a + b = b + a로 교환법칙이 성립합니다.
Ready to practice? Drill vector addition and the rest of linear algebra on C:Vector, or review the full vector reference and the related vector subtraction and scalar multiplication.