Vector Calculator
Add and subtract vectors, and find their magnitudes, dot and cross products, angle, unit vector and projection.
About the Vector Calculator
A vector has size and direction. The dot product measures how much two vectors point the same way (zero means perpendicular). The cross product of two 3D vectors is a third vector perpendicular to both, whose length equals the area of the parallelogram they span.
The formula
a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cos θ
- |a|
- √(a₁² + a₂² + a₃²)
- a × b
- (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)
- proj_b a
- (a·b ÷ |b|²) b
- [a b c]
- c · (a × b), the volume of the box on a, b, c
Worked example
a = (3, −2, 5), b = (1, 4, −1)
- a · b = 3 − 8 − 5 = −10
- a × b = (−18, 8, 14)
- With c = (2, 0, 1): [a b c] = c · (a × b) = −36 + 0 + 14 = −22
Questions people ask
What does a negative dot product mean?
The angle between the vectors is more than 90°.
Can I enter 2D vectors?
Yes; the cross product then gives the z-component, which is the signed area.