Dot Product Calculator
Calculate the dot product of two 3D vectors. Also shows the angle between them and scalar projection.
Dot
32.0000
Formula
## Dot Product ### Formula **A · B = a₁b₁ + a₂b₂ + a₃b₃** ### Geometric Interpretation **A · B = |A| × |B| × cos(theta)** So: **theta = arccos(A · B / (|A| × |B|))** ### Properties - If A · B = 0, the vectors are perpendicular - If A · B > 0, the angle is acute (less than 90°) - If A · B < 0, the angle is obtuse (greater than 90°)
Exemplo Resolvido
Dot product of (1,2,3) and (4,5,6).
- 01A · B = 1×4 + 2×5 + 3×6 = 4 + 10 + 18 = 32
- 02|A| = √(1+4+9) = √14 ≈ 3.742
- 03|B| = √(16+25+36) = √77 ≈ 8.775
- 04cos(theta) = 32/(3.742 × 8.775) ≈ 0.9746
- 05theta ≈ 12.93°
Perguntas Frequentes
What is the dot product?
The dot product is a way to multiply two vectors to get a scalar (single number). It measures how much two vectors point in the same direction.
When is the dot product zero?
The dot product is zero when the vectors are perpendicular (orthogonal). This is a common test for perpendicularity.
What is the difference between dot product and cross product?
The dot product gives a scalar and measures alignment. The cross product gives a vector perpendicular to both inputs and measures the area of the parallelogram they span.
Aprender