Dot Product

1. What Is the Dot Product?

The dot product (or scalar product) multiplies two vectors and returns a single number, not a vector. That number measures how much the two vectors point in the same direction. It is one of the most useful operations in all of mathematics — it powers length, angle, projection, and the entire idea of orthogonality.

Both vectors must have the same dimension. The output is always a scalar.

내적(스칼라곱)은 두 벡터를 곱해 하나의 수(스칼라)를 반환하며, 두 벡터가 얼마나 같은 방향을 향하는지를 측정합니다. 길이·각도·투영·직교의 토대가 되는 핵심 연산이며, 두 벡터의 차원은 같아야 합니다.

2. The Formula & a Worked Example

Multiply matching components and add the results:

a · b = a₁b₁ + a₂b₂ + … + aₙbₙ

Worked example. For a = (1, 2, 3) and b = (4, −5, 6):

a · b = (1)(4) + (2)(−5) + (3)(6) = 4 − 10 + 18 = 12

대응 성분끼리 곱한 뒤 모두 더합니다: a · b = a₁b₁ + … + aₙbₙ. 예: (1, 2, 3) · (4, −5, 6) = 4 − 10 + 18 = 12.

3. The Geometric Meaning

The dot product also equals the lengths times the cosine of the angle between the vectors:

a · b = |a| |b| cos θ

This rearranges to the angle formula cos θ = (a · b) / (|a| |b|), and the sign is an instant geometric test:

• a · b > 0 — angle is acute (vectors broadly agree).
• a · b = 0 — vectors are perpendicular (orthogonal).
• a · b < 0 — angle is obtuse (vectors broadly oppose).

내적은 a · b = |a| |b| cos θ 와도 같습니다. 따라서 cos θ = (a · b)/(|a| |b|)이며, 부호로 각을 판정합니다: 양수면 예각, 0이면 직교(수직), 음수면 둔각입니다.

4. Key Properties

• Commutative. a · b = b · a.
• Distributive. a · (b + c) = a · b + a · c.
• Scalar factoring. (k a) · b = k (a · b).
• Self dot = length squared. a · a = |a|². For a = (1, 2, 3), a · a = 1 + 4 + 9 = 14, so |a| = √14.
• Orthogonality test. Nonzero a and b are perpendicular exactly when a · b = 0.

교환법칙 a · b = b · a, 분배법칙 a · (b + c) = a · b + a · c, 스칼라 (k a) · b = k (a · b), 자기내적 a · a = |a|² (예: (1,2,3)의 경우 14 → |a| = √14), 직교 판정 a · b = 0 이 성립합니다.

5. Frequently Asked Questions

Is the dot product a vector or a scalar? A scalar — a single number. (The cross product is the operation that returns a vector.)

How do you know if two vectors are perpendicular? Compute the dot product; if it equals 0, the vectors are orthogonal.

How do you find the angle between two vectors? Rearrange the geometric formula: θ = arccos((a · b) / (|a| |b|)).

내적은 스칼라(수)입니다 — 벡터를 반환하는 것은 외적입니다. 내적이 0이면 두 벡터는 수직입니다. 각도는 θ = arccos((a · b)/(|a| |b|))로 구합니다.

Want a quick answer? The Dot Product Calculator computes a · b and each vector’s magnitude for your own vectors.

Ready to practice? Drill the dot product and the rest of linear algebra on C:Vector, or review the full vector reference and the related cross product, angle between vectors, and projection.

내 벡터로 바로 계산하려면 내적 계산기로 a · b와 크기를 즉시 구할 수 있습니다. 실전 연습은 C:Vector에서, 전체 개념은 벡터 레퍼런스와 외적, 벡터 사이 각, 투영 문서에서 이어집니다.