Vector Dot and Cross Product Calculator
Dot product, cross product and its magnitude, the angle between two vectors and their lengths.
Results
Dot product
32.000000
Cross product x
-3.000000
Cross product y
6.000000
Cross product z
-3.000000
Cross product magnitude
7.348469
Angle
12.93315
°
Apparent magnitude
3.741657
Apparent magnitude
8.774964
What this tool does
The dot product is a number and measures how much the vectors point the same way — zero means perpendicular, negative means more than a right angle apart. The cross product is a vector at right angles to both, and its length equals the area of the parallelogram they span, which is why it appears in torque, in magnetic force and in surface normals.
Formula
a·b = axbx + ayby + azbz ; |a×b| = |a||b| sin θ
Variables
| Symbol | Meaning | Unit |
|---|---|---|
ax | a (x) | — |
ay | a (y) | — |
az | a (z) | — |
bx | b (x) | — |
by | b (y) | — |
bz | b (z) | — |
DP | Dot product | — |
CX | Cross product x | — |
CY | Cross product y | — |
CZ | Cross product z | — |
CM | Cross product magnitude | — |
AN | Angle | ° |
MA | Apparent magnitude | — |
MB | Apparent magnitude | — |
Worked example
- a (x)1
- a (y)2
- a (z)3
- b (x)4
- b (y)5
- b (z)6
- Dot product32.000000
- Cross product x-3.000000
- Cross product y6.000000
- Cross product z-3.000000
- Cross product magnitude7.348469
- Angle12.93315 °
- Apparent magnitude3.741657
- Apparent magnitude8.774964
Limitations
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.