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The vector object

The vector object is not a displayable object but is a powerful aid to 3D computations. Its properties are similar to vectors used in science and engineering.

vec(x,y,z)

This creates a 3D vector object with the given components x, y, and z.

Vectors can be added or subtracted from each other, or multiplied by an ordinary number. For example,

var v1 = vec(1,2,3)
var v2 = vec(10,20,30)
v1+v2 # same as vec( 1, 22, 33)
20*v1 # same as vec(20, 40, 60)

You can refer to individual components of a vector:

v2.x is 10, v2.y is 20, v2.z is 30

It is okay to make a vector from a vector: vec(v2) is still vec(10,20,30). This is a convenient way to make a separate copy of a vector.

Vector functions

The following functions are available for working with vectors:

mag(A) = A.mag = |A|, the magnitude of a vector

mag2(A) = A.mag2 = |A|*|A|, the vector's magnitude squared

norm(A) = A.norm() = A/|A|, a unit vector in the direction of the vector

hat(A) = A.hat = A/|A|, a unit vector in the direction of the vector; an alternative to A.norm(), based on the fact that unit vectors are customarily written in the form ĉ, with a "hat" over the vector

dot(A,B) = A.dot(B) = A dot B, the scalar dot product between two vectors

cross(A,B) = A.cross(B), the vector cross product between two vectors

diff_angle(A,B) = A.diff_angle(B), the angle between two vectors, in radians

proj(A,B) = A.proj(B) = dot(A,norm(B))*norm(B), the vector projection of A along B

comp(A,B) = A.comp(B) = dot(A,norm(B)), the scalar projection of A along B

A.equals(B) is True if A and B have the same components (which means that they have the same magnitude and the same direction)

vec.random() produces a vector each of whose components are random numbers in the range -1 to +1

Some examples:

mag(A) # calculates length of A
mag(vec(1,1,1))  # = sqrt(3) = 1.732...
mag2(vec(1,1,1)) # = 3; magnitude squared

It is possible to reset the magnitude or the magnitude squared of a vector:

v2.mag = 5 # sets magnitude to 5;
# no change in direction
v2.mag2 = 2.7 # sets squared magnitude
# of v2 to 2.7

You can reset the magnitude to 1 with norm():

norm(A) # A/|A|, normalized; magnitude 1
norm(vec(1,1,1)) # vec(1,1,1)/sqrt(3)

You can also write v1.norm() or v1.hat. For convenience, norm(vec(0,0,0)) is calculated to be vec(0,0,0).

You can change the direction of a vector without changing its magnitude:

v2.hat = v1 # changes the direction of v2
# but not its magnitude

To calculate the angle between two vectors (the "difference" of the angles of the two vectors):

ang = v1.diff_angle(v2)

You can also write v1.diff_angle(v2). For convenience, if either of the vectors has zero magnitude, the difference of the angles is calculated to be zero.

cross(A,B) or A.cross(B) gives the cross product of two vectors, a vector perpendicular to the plane defined by A and B, in a direction defined by the right-hand rule: if the fingers of the right hand bend from A toward B, the thumb points in the direction of the cross product. The magnitude of this vector is equal to mag(A)*mag(B)*sin(A.diff_angle(B)).

dot(A,B) or A.dot(B) gives the dot product of two vectors, which is an ordinary number equal to mag(A)*mag(B)*cos(A.diff_angle(B)). If the two vectors are normalized, the dot product gives the cosine of the angle between the vectors, which is often useful.

Rotating a vector

There is a function for rotating a vector:

v2 = rotate(v1, angle=a, axis=vec(x,y,z))

The angle must be in radians. The default axis is (0,0,1), for a rotation counterclockwise in the xy plane around the z axis. There is no origin for rotating a vector. You can also write v2 = v1.rotate(angle=theta, axis=vec(1,1,1)). There is also a rotate capability for objects.

The JavaScript versions are v2 = rotate(v1, {angle:a, axis=vec(x,y,z}) and v2 = v1.rotate({angle:a, axis=vec(x,y,z}).

There are functions for converting between degrees and radians, where there are 2*pi radians in 360 degrees: