Dot Product Calculator
a · b and vector magnitudes · 내적 계산기
Enter two vectors of the same length (2D, 3D, or higher) to get a · b and each magnitude.
How to use this calculator
- Enter the components of vector a, comma-separated (e.g. 1, 2, 3).
- Enter vector b with the same number of components.
- Press Calculate for a · b plus the magnitudes |a| and |b|.
벡터 a의 성분을 쉼표로 구분해 입력하고(예: 1, 2, 3), 같은 개수의 성분으로 벡터 b를 입력한 뒤 Calculate를 누르면 a·b와 크기 |a|·|b|가 나옵니다.
What the dot product is
The dot product multiplies matching components and sums them: a · b = a₁b₁ + a₂b₂ + … + aₙbₙ — a single number. Geometrically a · b = |a| |b| cos θ, so it reveals the angle between the vectors and is zero exactly when they are perpendicular. For the full treatment see the dot product guide, magnitude, and the angle between vectors.
내적은 같은 성분끼리 곱해 더한 하나의 수입니다: a · b = a₁b₁ + a₂b₂ + … + aₙbₙ. 기하적으로 a · b = |a||b|cos θ라서 두 벡터의 각도를 알려주고, 수직일 때 정확히 0입니다. 자세한 내용은 내적·크기·사잇각 문서를 참고하세요.
Worked example
For a = (1, 2, 3) and b = (4, 5, 6): a · b = 1·4 + 2·5 + 3·6 = 4 + 10 + 18 = 32. The magnitudes are |a| = √14 ≈ 3.742 and |b| = √77 ≈ 8.775. Press Try an example above to load these vectors.
a = (1, 2, 3), b = (4, 5, 6): a·b = 4 + 10 + 18 = 32. 크기는 |a| = √14 ≈ 3.742, |b| = √77 ≈ 8.775.
Frequently asked questions
How do you calculate a dot product? Multiply each pair of matching components and add the results: a · b = a₁b₁ + a₂b₂ + … + aₙbₙ.
Does it work for 2D and 3D? Yes — any two vectors with the same number of components, in any dimension.
What does a dot product of 0 mean? The vectors are perpendicular (orthogonal), since a · b = |a| |b| cos 90° = 0.
내적은 같은 성분끼리 곱해 더합니다(a₁b₁ + a₂b₂ + …). 차원에 상관없이 성분 개수가 같으면 됩니다. 내적이 0이면 두 벡터는 수직입니다.
Keep practicing
Drill vector operations on C:Vector, read the dot product and magnitude guides, or browse more tools on the calculators hub.