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`A = 1/2 ⋅ |V| ⋅ |U| ⋅ sin(α) `

Enter a value for all fields

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**): x, y and z components of vector V separated by commas (e.g. 4,5,9) - (
**U**): x, y and z components of vector U separated by commas (e.g. 4,5,9)

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

- Compute the length (magnitude) of both vectors. They represent the length of two legs of a triangle (|V| and |U|).
- Compute the angle between the vectors(α).
- 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.

**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.**(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- Vector Normal to a Plane Defined by Three Points

- Area of a triangle base on the base and height
- Area of a triangle using two sides and the interior angle.
- Area of a triangle using two angles and the interior side.
- Area of a triangle using three known sides.
- Area of a triangle defined by three points (x,y) in a plane.
- Height of a triangle based on the area and base
- Interior Angle of a triangle based on the length of three sides
- Semi-perimeter of a triangle
- Area between two vectors