Vector Components 3D

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