免费点积计算器
计算两个向量的点积(内积/数量积)。
Dot
32.0000
公式
## 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°)
计算示例
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°
常见问题
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.
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