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.

vector(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,

v1 = vector(1,2,3)
v2 = vector(10,20,30)
print(v1+v2) # displays <1 22 33>
print(2*v1) # displays <2 4 6>

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: vector(v2) is still vector(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

For convenience, norm(vec(0,0,0)) or vec(0,0,0).hat is calculated to be vec(0,0,0).

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)

vector.random() produces a vector each of whose components is a random number in the range -1 to +1

Some examples:

mag(A) # calculates length of A
mag(vector(1,1,1)) # = sqrt(3) = 1.732...
mag2(vector(1,1,1)) # = 3, the 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 of 1
norm(vector(1,1,1)) = vector(1,1,1)/sqrt(3)

You can also write v1.norm() or v1.hat.

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

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

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

diff_angle(v1,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 mag(A)*mag(B)*sin(diff_angle(A,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(diff_angle(A,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=vector(x,y,z))

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

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

radians(360) is equivalent to 2*pi

degrees(2*pi) is equivalent to 360