Vector Calculator

The Vector Calculator is provided in support of our Physics Tutorials on Vectors and Scalars which explores addition and subtraction of vectors, multiplication of a vector by a scalar, dot (scalar) product of two vectors and the vector product of two vectors with practical working examples and formula. A list of the supporting Vector and Scalar Physics Tutorials is available at the bottom of this page. This calculator will calculate:

  1. The magnitudes of each vectors when the coordinates are given
  2. The sum of two vectors
  3. The difference of two vectors
  4. The dot product of two vectors
  5. The magnitude of the cross product of two vectors
  6. The angle between two vectors

These results will be used to find the following physical quantities:

  1. The resultant of two vector quantities
  2. Work done by a force when it moves an object at a certain linear distance
  3. Moment of a force
  4. The angle between a Force and Linear distance a system moves linearly or rotates around a fixed point (pivot)

Vector Calculator
Vector Calculator Results (detailed calculations and formula below)
Magnitude of Vector u (|u|)
Magnitude of Vector v (|v|)
Sum of Vectors u and v (u+v)
Difference of Vectors u and v (u-v)
Dot Product of Vectors u and v (u∙v)
Magnitude of the Cross Product of Vectors u and v (|u∙v|)
Cosine of Angle Formula and Calculation (cos θ)
Sine of Angle Formula and Calculation (sin θ)
Tan of Angle Formula and Calculation (tan θ)
Angle θ (angle θ)

Please note that the formula for each calculation along with detailed calculations is shown further below this page. As you enter the specific factors of each vector calcualtion, the Vector Calculator will automatically calculate the results and update the formula elements with each element of the vector calculation. You can then email or print this vector calculation as required for later use.

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Magnitude of Vector u Formula and Calculation

|u| = √u2x + u2y + u2z
|u| = √2 + 2 + 2
|u| = √ + +
|u| =

Magnitude of Vector v Formula and Calculation

|v| = √v2x + v2y + v2z
|v| = √2 + 2 + 2
|v| = √ + +
|v| =

Sum of Vectors u and v Formula and Calculation

u + v = ux + uy, uz + vx, vy + vz
u + v = + , + , +
u + v =

Difference of Vectors u and v Formula and Calculation

u - v = ux - uy, uz - vx, vy - vz
u - v = - , - , -
u - v =

Dot Product of Vectors u and v Formula and Calculation

u∙v = (ux∙vx) + (uy∙vy) + (uz∙vz)
u∙v = () + () + ()
u∙v =

Magnitude of the Cross Product of Vectors u and v Formula and Calculation

|u∙v|x = uy ∙ vz - uz ∙ vy
|u∙v|x = -
|u∙v|x =
|u∙v|y = uz ∙ vx - ux ∙ vy
|u∙v|y = -
|u∙v|y =
|u∙v|z = ux ∙ vy - uy ∙ vx
|u∙v|z = -
|u∙v|z =
|u∙v| = √/|u∙v|2x + |u∙v|2y + |u∙v|2z
|u∙v| = √/ + +
|u∙v| =

Cosine of Angle Formula and Calculation

cos θ = u ∙ v/|u| ∙ |v|
cos θ = /
cos θ =

Sine of Angle Formula and Calculation

sin θ = |u ∙ v|/|u| ∙ |v|
sin θ = /
sin θ =

Tan of Angle Formula and Calculation

tan θ = sin θ/cos θ
tan θ = /
tan θ =

Angle θ Formula and Calculation

angle θ = arctan (tan θ)
angle θ = arctan ()
angle θ = ° (degrees)

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