Vectors¶
Vectors are an ordered array of numbers in ℝ. All vectors have
four components X, Y, Z, and W and are thus in ℝ4,
but often the W component is equal to zero or one. Thus
they usually they represent a point or direction in a 3-D space.
Points and directions¶
A Point is a vector with the W component equal to one
and a Direction is a vector with the W component equal to zero.
Arithmetic¶
The following binary operators can be used on vectors: +, -, and *.
The * can also operate on a vector and a single element.
Unary operators that operate on a single vector are: - and abs.
Summary
Vector addition and multiplication is associative and commutative:
-
A + (B + C) = (A + B) + C -
A + B = B + A
Similar for (element-wise) multiplication.
Points and directions¶
Subtracting two Points results in a Direction.
Adding a Direction to a Point gives another Point.
Subtracing or adding two Directions gives another Direction.
A Direction can also be scaled by a single element.
A Point cannot be scaled by an arbitrary element, but can be mirrored
using the unary - operator.
Dot and cross products¶
The function Dot returns the dot or inner product of two vectors.
It is the sum of the product of the corresponding elements in the vectors.
It can also be expressed in terms of the norm (magnitude) and angle:
Dot (A, B) = Norm (A) * Norm (B) * Angle (A, B)
The dot product of two unit vectors represents the angle between them and is 0 if they are orthogonal, 1 if they are in the same direction, and -1 if they are in the opposite direction.
The dot product a ∙ b is sometimes written as aT b
The cross product of two vectors can be computed with the function Cross.
Summary
Dot product is commutative: Dot (A, B) = Dot (B, A).
The cross product is anti-commutative: ::ada Cross (A, B) = Cross (-B, A).
Length and normalization¶
The functions Magnitude and Norm return a scalar that represents
the magnitude of a vector; the distance between a point in a space and
the origin.
A unit vector is a vector with a magnitude of one. The function Normalized
will return True if a vector is a unit vector and False if the magnitude
is not equal to one.
The functions Normalize and Normalize_Fast will normalize the given vector.
Thus the magnitude of the vector returned by these functions is one. Normalize_Fast
may use SIMD intrinsics for single precision floating-point vectors, which will
be faster, but also less accurate than the implementation used by Normalize.
Distance and angle¶
The function Distance returns the distance between two Points and the
function Angle returns the angle in radians between two vectors.
The angle between two unit vectors is equal to the dot product of the two vectors.
Projection¶
The function Projection returns the projection of a vector on some other vector.
The magnitude of the projected vector is equal to Dot (U, V) / Magnitude2 (V)
The function Perpendicular returns a vector perpendicular to the projection
of the vector on the second vector.
For example, if W = Projection (U, V), then U - W = Perpendicular (U, V)
and (U - W) + W = U.
It can also be said that the dot product of the vectors U - W and W is zero
and thus they are orthogonal.
Interpolation¶
The function Slerp returns the interpolated unit vector on the shortest arc
between two vectors. This can be useful for animations.