免费点积计算器

计算两个向量的点积(内积/数量积)。

Vector A: x

Vector A: y

Vector A: z

Vector B: x

Vector B: y

Vector B: z

Dot

32.0000

Mag A3.7417

Mag B8.7750

Angle12.9332 deg

查看公式 →

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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