Vector Geometry

Vectors can be used to express shapes and objects. These have geometry that can be expressed in terms of the vectors themselves.

Unit Triangle Formed by a Vector

          a
          *
  ||a||  /|
        / |
       /  | ||a|| sin(θ)
      /   |
     /θ   |
    /_____|

       ||a|| cos(θ)

Angle Formed by Two Vectors

Given two sides of a triangle, the angle between them is characterized by the cosine function:

cos(θ) = [ adjacent ] / [ hypotenuse ]

For any two given vectors, a similar formula can be used. To force a right triangle however, the cosine must be calculated with respect to the hypotenuse and the projection of the hypotenuse on bthe other angle.

cos(θ) = [ (a⃗ · b⃗)/||b⃗|| ] / [ ||a⃗|| ]

cos(θ) = (a⃗ · b⃗)/(||a⃗|| ||b⃗||)

θ = cos^-1((a⃗ · b⃗)/(||a⃗|| ||b⃗||))

Parallelogram

The area of a parallelogram formed by two vectors can be calculated in several ways.

1. ||a⃗|| ||b⃗|| sin(θ) where θ is the angle formed by the two vectors

   • Area is equal to base times height and height is clearly given by ||b⃗|| sin(θ)

2. a⃗' · b⃗ where a⃗' is a⃗ rotated by 90 degrees (or π/2 radians) counter-clockwise

   • This rotation 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 rotation can also be expressed with a rotation matrix.

   • This form is derived by forcing the first 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

3. det(a⃗, b⃗)

   • The second form expands to a₁b₂ - a₂b₁. This is definitionally the determinant of the two vectors.

4. ||a × b||

   • The distance of the cross product also gives the area

Note that these calculations can return a positive or negative number, so the sign must be discarded to treat it as geometric area.

Finally, note that the #2 and #3 can trivially be seen to not hold in higher dimensions; the matrix would be rectangular and only square matrices are invertible. #4 does in fact hold in that circumstance.

Triangles

As above, multiplied by 1/2.

Plane Formed by Two Vectors

To find the equation of a plane described by two vectors, first calculate the normal vector using the cross product of the vectors.

If the plane passes through the origin, simply extract the normal vectors coefficients and set to zero: i.e. n⃗₁x + n⃗₂y + n⃗₃z = 0.

More generally, the equation is constructed given a point P = [a b c] on the plane: i.e. n⃗₁(x-a) + n⃗₂(y-b) + n⃗₃(z-c) = 0

Angle Formed by Two Planes

Following from the above, the angle formed by two planes is given by:

cos(θ) = |(n₁ · n₂)|/(||n₁|| ||n₂||)

θ = cos^-1(|(n₁ · n₂)|/(||n₁|| ||n₂||))

where n_i is the normal vector for plane i. Recall that the trivial normal vector of a plane defined like 3x + 5y - z = 4 is [3 5 -1].

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.

Cylinder

The standard form equation of a cylinder which stretches infinitely along the z-axis is (x-a)² + (y-b)² = r² where the point P = [a b z] is the center of the cylinder and r is the radius.

Sphere

The standard form equation of a sphere is (x-a)² + (y-b)² + (z-c)² = r² where the point P = [a b c] is the center of the sphere and r is the radius.

CategoryRicottone