Orthogonality
Orthogonality is a generalization of perpendicularity.
Description
The notation for orthogonality is ⊥, as in a⃗ ⊥ b⃗.
In R3 space, a vector is orthogonal to a plane.
Dot Product
If two vectors are perpendicular, they must satisfy the Pythagorean theorem. It follows that the dot product must be 0.
aTa + bTb = (a+b)T(a+b)
aTa + bTb = aTa + bTb + aTb + bTa
0 = aTb + bTa
0 = 2(aTb)
0 = aTb
Normal Vectors
A frequent operation in vector calculus is identifying a normal vector. These are usually notated as N, or n if a unit normal vector.
To identify a vector normal to a plane formed by two vectors, use the cross product on those vectors.
To identify a vector normal to a plane formed by three points (A, B, and C), calculate the tangent vectors as B-A and C-A. Then use the above strategy.
For a given plane (ax + by + cz = d), the trivial normal vector is [a b c].
In all cases, it should be understood that there are infinitely many normal vectors. For any given vector is normal to a given plane, it must be a similar to the trivial normal vector. That is, the given vector must be a scaled version of the trivial normal vector.
Normal Planes
To identify a plane normal to a given vector a⃗, use the dot product to design a system of equations.
If the origin should be included in the solution, then the plane must be composed of a vector b⃗ from the origin to a point at (x, y, z). b⃗ is then characterized by [(x-0) (y-0) (z-0)] or [x y z]. The dot product of this and a⃗ must be 0. If the given vector a⃗ is [1 5 10] then:
a⃗ · b⃗ = 0
[1 5 10] · [x y z] = 0
x + 5y + 10z = 0
The plane is characterized by this equation.
If a given point P should be included in the solution, as opposed to the origin, simply update with [(x-p1) (y-p2) (z-p3)]. If the given point P is (2, 1, -1) then:
a⃗ · b⃗' = 0
[1 5 10] · [(x-2) (y-1) (z+1)] = 0
(x-2) + 5(y-1) + 10(z+1) = 0
x + 5y + 10z = -3
This system reveals a parallel plane with a constant offset of -3.