Area between Vectors

The Vector Area calculator Vectors U and V in three dimensions computes the area swept between two vectors (V and U) in Euclidean three dimensional space. This is the light blue area in the graphic.

INSTRUCTIONS: Enter the following in meters:

• (V):  Vector V in meters
• (U):  Vector U in meters

Area between Vectors (A): The calculator returns area in square meters.  However, this can be automatically converted to compatible units via the pull-down menu.

The Math / Science

The Area between two Vectors (A) calculator computes the two dimensional area between two 3D vectors.  The formula to compute the area is:

1. Compute the length (magnitude) of both vectors.  They represent the length of two legs of a triangle (|V| and |U|).
2. Compute the angle between the vectors(α).  
3. Use the two lengths of and the angle to compute the area of the triangle, where:

       A = 1/2 ⋅ |V| ⋅ |U| ⋅ sin(α) 

This formula lets the user enter two three-dimensional vectors (V and U)  with X, Y and Z components.   Note the dot product of two unit vectors is equal to the cosine of the angle between the two vectors.

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

A triangle is a polygon with three sides, three vertices (corners), and three angles. Triangles can be classified based on the lengths of their sides and the measures of their angles as follows:

By Side Lengths:

• Equilateral Triangle: All three sides are equal in length.
• Isosceles Triangle: Two sides are equal in length.
• Scalene Triangle: All three sides have different lengths.

By Angle Measures:

• Acute Triangle: All three angles are less than 90 degrees.
• Right Triangle: One angle is exactly 90 degrees.
• Obtuse Triangle: One angle is greater than 90 degrees.

The sum of the interior angles of any triangle always adds up to 180 degrees. 

Triangle Calculators

• Area of Triangle (base and height)
• Area of Triangle (two sides and interior angle)
• Area of Triangle (two angles and interior side)
• Area of Triangle (three sides)
• Area of Equilateral Triangle
• Area of Triangle (three points)
• Height of Triangle
• Width of Triangle
• Triangle Perimeter
• Interior Angle of a triangle based on the length of three sides
• Semi-perimeter of a triangle
• Area of Circle Within a Triangle
• Area of Circle Around a Triangle
• Area between two vectors
• Triangle Volume