Vector Components 3D

` = "Vector Components 3D"`

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The Vector Components (3D) calculator computes the components of a vector in three dimensions (3D).

INSTRUCTIONS: Enter the following:

(`vecF`) Vector F

Vector Components: The calculator returns the following:

|`vecF`| - Magnitude of Vector (Note: Units will be the same as input vector)
`hatF` - Unit Vector
(ρ,θ,φ) Spherical Coordinates with θ and φ in degrees (θ°,φ°)
(r,Θ,z) Cylindrical Coordinates with θ in degrees (θ°)
α - Angle between `vecF` and x axis
φ - Angle between `vecF` and y axis
θ - Angle between `vecF` and z axis

The Math / Science

To compute the angle between the vector and axes, the unit vector is computed.

`hatF` = `vecF / |vecF|`

Then compute the dot products between the unit vector `hatF` and the unit vectors for the axes (1,0,0) for x, (0,1,0) for y and (0,0,1) for z. The arccosine of each dot product is the angle between them in radians.

α = acos( `hatF * 1,0,0 `)
φ = acos( `hatF * 0,1,0 `)
θ = acos( `hatF * 0,0,1 `)

3D Vector Functions

k⋅V - scalar multiplication
V/k - scalar division
V / |V| - Computes the Unit Vector
|V| - Computes the magnitude of a vector
U + V - Vector addition
U - V - Vector subtraction
|U - V| - Distance between vector endpoints.
|U + V| - Magnitude of vector sum.
V • U - Computes the dot product of two vectors
V x U - Computes the cross product of two vectors
V x U • W - Computes the mixed product of three vectors
Vector Angle - Computes the angle between two vectors
Vector Area - Computes the area between two vectors
Vector Projection - Compute the vector projection of V onto U.
Vector Rotation - Compute the result vector after rotating around an axis.
Vector Components 3D - Returns a vector's magnitude, unit vector, spherical coordinates, cylindrical coordinates and angle from each axis.
(ρ, θ, φ) to (x,y,z) - Spherical to Cartesian coordinates
(x,y,z) to (ρ, θ, φ) - Cartesian to Spherical coordinates
(r, θ, z) to (x,y,z) - Cylindrical to Cartesian coordinates
(x,y,z) to (r, θ, z) - Cartesian to Cylindrical coordinates
(x,y) to (r, θ) - Cartesian to Polar
(r, θ) to (x,y) - Polar to Cartesian
Vector Normal to a Plane Defined by Three Points