Vector Calculator
Magnitudes, dot product, and angle for two 2D vectors.
What it does
Vectors carry size with direction: (3,4) spans 5 units while (-4,3) spans 5 the other way, and their dot product of zero proves them perpendicular at 90 degrees. Magnitude, dot, and angle answer every 2D vector question between them.
The tool above takes both component pairs. It returns magnitudes with dot product and enclosed angle beside them. The default (3,4) with (-4,3) gives 5 and 5 with dot 0 at 90 deg.
Methods match OpenStax with HyperPhysics beside them: head-to-tail addition sums components, Pythagoras roots magnitudes, and arccosine normalizes the dot into degrees. Worked checks agree, from 9 east plus 5 north at 10.3 and 29.1 degrees to zero vectors holding no angle.
Dot sign reads alignment at a glance: positive runs together, zero stands square, negative pulls apart. Parallel twins share 0 degrees while antiparallel pairs take 180.
How to use
- Type A x and A y.
- Type B x and B y.
- Read magnitudes first, dot beside them.
- Read enclosed angle last.
- Zero dot means perpendicular pair.
- Zero vector means angle undefined.
Formula
- |A|
- Vector magnitude from Pythagoras.
- dot
- Componentwise sum measuring alignment.
- angle
- Enclosed degrees, 0 to 180.
Units ride along untouched; mixing meters with seconds still computes but means nothing. Arccosine clamps protect rounding at parallels.
Worked example
Example: (3,4) and (-4,3)
Defaults: A (3,4), B (-4,3).
- |A|: root(9 + 16) = 5.
- |B|: root(16 + 9) = 5.
- Dot: 3 x -4 + 4 x 3 = 0.
- Angle: arccos(0 / 25) = 90 deg.
- Read: perpendicular pair confirmed.
- Tool shows 5, 5, 0, 90 deg.
Reference table
| Vectors | Magnitudes | Dot and angle |
|---|---|---|
| (3,4), (-4,3) | 5 and 5 | 0 at 90 deg |
| (1,0), (0,1) | 1 and 1 | 0 at 90 deg |
| (2,0), (3,0) | 2 and 3 | 6 at 0 deg |
| (9,5), (0,0) | 10.3 and 0 | 0, angle undefined |
| (1,2), (3,4) | 2.24 and 5 | 11 |
Limits
- 2D only; 3D needs z terms.
- Zero vector holds no angle.
- Degrees out; radians need converting.
- Units must match per vector.
Common mistakes
- Adding magnitudes instead of components.
- Reading dot units as an angle.
- Forgetting arccosine domain clamps.
- Mixing unit systems mid vector.
- Assuming zero dot with zero vector meaning.
Try it yourself
Quick checks. Answers sit tucked under each question.
(1,0) and (0,1)?
1 and 1, dot 0, 90 deg.
(2,0) and (3,0)?
2 and 3, dot 6, 0 deg.
Magnitude of (9,5)?
10.3 with zero partner undefined.
Questions
Default answer?
5 and 5 with dot 0 at 90 deg.
Dot zero meaning?
Perpendicular vectors.
Negative dot?
Obtuse alignment past 90 degrees.
Zero vector?
Magnitude 0 with undefined angle.
Distance page versus this?
That measures between points; this grades arrows.
Cross product?
Out of 2D scope; needs the third dimension.