Library / Foundations: Maths / FMA-40
Foundations: Maths
Dot product
Scalar product of two vectors.
- Ref
- FMA-40
- Rev
- v1
- Field
- Foundations: Maths
- Result
- a·b
|a| and |b| are the lengths of the two vectors and θ the angle between them when drawn from the same point, from 0° (same direction) to 180° (opposite). The result is a number, not a vector. In components, a·b = aₓbₓ + a_yb_y + a_zb_z.
Scalar product of two vectors. It is largest when they point the same way and zero when they are at right angles.
Calculator
- Component of a along b2.5
- Component of a perpendicular to b4.33
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The dot product is zero for perpendicular vectors, so it tests for a right angle.
- Work is a dot product: a 5 N force along a 4 m path at 60° does 10 J, because only the component along the path counts.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| |b| Magnitude of b | 0.1 to 100 | 0.25 | 250 | 249.8 | +10% | +10 % |
| |a| Magnitude of a | 0.1 to 100 | 0.2 | 200 | 199.8 | +10% | +10 % |
| θ Angle between them | 0 to 180 ° | −20 | 20 | 40 | +10% | −18.7 % |
Sweep of Angle between them
| Angle between them (°) | Dot product |
|---|---|
| 0 | 20 |
| 22.5 | 18.48 |
| 45 | 14.14 |
| 67.5 | 7.654 |
| 90 | 1.225e-15 |
| 112.5 | −7.654 |
| 135 | −14.14 |
| 157.5 | −18.48 |
| 180 | −20 |
Sweep one variable
Variables and how each one contributes
Dot product
What the equation gives you. Everything else feeds it.
Magnitude of a
The length of the first vector. Directly proportional.
Magnitude of b
The length of the second vector. Directly proportional.
Angle between them
The angle between the vectors. Acts through cos θ: flat near 0° and steepest near 90°. From 60° to 70° the product falls from 10 to 6.8, and beyond 90° it turns negative.
Worked example with the values above
- cos 60° = 0.5.
- Product of the lengths: 5 × 4 = 20.
- a·b = 20 × 0.5 = 10.
Assumptions and limits
- Both vectors are measured from the same point or are free vectors.
- The angle is the smaller one between the directions, from 0° to 180°.
Common mistakes
- Using the angle between a vector and a plane instead of between the two vectors.
- Setting the calculator to radians when θ is in degrees.
Statistics and uncertainty
Near 90° the dot product is a small difference of large contributions, so a small error in the angle gives a large relative error in the result; near 0° or 180° the same angle error matters very little.
Where it is used
- Work done by a force along a displacement.
- Component of a velocity or wind along a line of sight.
Maths you need
- Vectors: magnitude and direction
- Cosine: the component along a direction