点积计算器 — 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).

  1. A · B = 1×4 + 2×5 + 3×6 = 4 + 10 + 18 = 32
  2. |A| = √(1+4+9) = √14 ≈ 3.742
  3. |B| = √(16+25+36) = √77 ≈ 8.775
  4. cos(theta) = 32/(3.742 × 8.775) ≈ 0.9746
  5. theta ≈ 12.93°