Vector Geometry
Vectors can be used to express shapes and objects. These have geometry that can be expressed in terms of the vectors themselves.
Contents
Parallelogram in the Cartesian Plane
The area of a parallelogram formed by two vectors can be calculated as ||a⃗|| ||b⃗|| sin(θ) = a⃗' · b⃗.
The first form should be self-explanatory. θ is the angle formed by the two vectors; its area is equal to base times height; its height is given by ||b⃗|| sin(θ).
The second form is in terms of a⃗', which is a⃗ rotated by 90 degrees (or π/2 radians) counter-clockwise.
This can be thought of as transforming a vector composed as [a1 a2] into [-a2 a1], i.e. flipping the sign of the second component and then swapping the components.
This can also be expressed with a rotation matrix.
The derivation for the second form is simply forcing it into the 'shape' of the dot product:
||a⃗|| ||b⃗|| sin(θ)
||a⃗'|| ||b⃗|| cos(θ') where θ' is the complement of θ, and is the angle formed by a⃗' and b⃗.
a⃗' · b⃗
This dot product expands out to a1b2 - a2b1. This is called the determinant of the two vectors, and notated as det(a⃗, b⃗).
Finally, note that these calculations can return a positive or negative number, so the sign must be discarded to treat it as geometric area.
Triangles
As above, multiplied by 1/2.
Parallelogram in Space
The area of a parallelogram formed by two three-dimensional vectors is given by the distance of the cross product. Recall that the cross product returns a vector. This product describes the parallelogram in terms of the unit vectors.
The determinant-based approach for calculating area of a parallelogram in the Cartesian plane can trivially be seen to not hold in higher dimensions; the matrix would be rectangular and only square matrices are invertible.
Parallelepiped
The volume of a parallelepiped formed by three vectors is equal to the determinant of the three vectors, and notated as det(a⃗, b⃗, c⃗). Recall that the determinant of a matrix does not change with transposition, so this 3 by 3 matrix can be constructed either of columns or rows.
The general approach for calculating this by hand begins with separating the first row (i.e., [a1 b1 c1]). The determinant of a 3 by 3 matrix is a linear combination of the determinants of the 2 by 2 matrices according to the 3rd row.
This can also be reformulated in terms of the dot and cross products as a⃗ · (b⃗ ⨯ c⃗). This is called the triple product.
If a⃗ exists in the plane formed by b⃗ and c⃗, then they in fact form a parallelogram with 0 volume. This matches with the first definition: the determinant is 0. This matches with the second definition as well. Recall that the cross product of b⃗ and c⃗ returns a vector that is orthogonal to both. If a⃗ exists on that plane, then the cross product is orthogonal to all of a⃗, b⃗, and c⃗. The dot product of two orthogonal vectors is always 0.