Vector Calculus 3d Motion

## Description

This interact illustrates the relationships between common vectors in vector calculus in 3d. The interface is exactly the same as the interface of Vector Calculus: 2-D Motion.

## Sage Cell

#### Code

```
# 3-D motion and vector calculus
# Copyright 2009, Robert A. Beezer
# Creative Commons BY-SA 3.0 US
#
#
# 2009/02/15 Built on Sage 3.3.rc0
# 2009/02/17 Improvements from Jason Grout
#
# variable parameter is t
# later at a particular value named t0
#
# un-comment double hash (##) to get
# time-consuming torsion computation
#
var('t')
assume(t, 'real')
#
# parameter range
#
start=-4*pi
stop=8*pi
#
# position vector definition
# edit here for new example
# example is wide ellipse
# adjust figsize in final show() to get accurate aspect ratio
#
a=1/(8*pi)
c=(3/2)*a
position=vector( (exp(a*t)*cos(t), exp(a*t)*sin(t), exp(c*t)) )
#
# graphic of the motion itself
#
path = parametric_plot3d( position.list(), (t, start, stop), color = "black" )
#
# derivatives of motion, lengths, unit vectors, etc
#
velocity = derivative( position, t)
acceleration = derivative(velocity, t)
speed = velocity.norm()
speed_deriv = derivative(speed, t)
tangent = (1/speed)*velocity
dT = derivative(tangent, t)
normal = (1/dT.norm())*dT
binormal = tangent.cross_product(normal)
## dB = derivative(binormal, t)
#
# interact section
# slider for parameter, 24 settings
# checkboxes for various vector displays
# computations at one value of parameter, t0
#
@interact
def _(t0 = slider(float(start), float(stop), float((stop-start)/24), float(start) , label = "Parameter"),
pos_check = ("Position", True),
vel_check = ("Velocity", False),
tan_check = ("Unit Tangent", False),
nor_check = ("Unit Normal", False),
bin_check = ("Unit Binormal", False),
acc_check = ("Acceleration", False),
tancomp_check = ("Tangential Component", False),
norcomp_check = ("Normal Component", False)
):
#
# location of interest
#
pos_tzero = position(t=t0)
#
# various scalar quantities at point
#
speed_component = speed(t=t0)
tangent_component = speed_deriv(t=t0)
normal_component = sqrt( acceleration(t=t0).norm()^2 - tangent_component^2 )
curvature = normal_component/speed_component^2
## torsion = (1/speed_component)*(dB(t0)).dot_product(normal(t0))
#
# various vectors, mostly as arrows from the point
#
pos = arrow3d((0,0,0), pos_tzero, rgbcolor=(0,0,0))
tan = arrow3d(pos_tzero, pos_tzero + tangent(t=t0), rgbcolor=(0,1,0) )
vel = arrow3d(pos_tzero, pos_tzero + velocity(t=t0), rgbcolor=(0,0.5,0))
nor = arrow3d(pos_tzero, pos_tzero + normal(t=t0), rgbcolor=(0.5,0,0))
bin = arrow3d(pos_tzero, pos_tzero + binormal(t=t0), rgbcolor=(0,0,0.5))
acc = arrow3d(pos_tzero, pos_tzero + acceleration(t=t0), rgbcolor=(1,0,1))
tancomp = arrow3d(pos_tzero, pos_tzero + tangent_component*tangent(t=t0), rgbcolor=(1,0,1) )
norcomp = arrow3d(pos_tzero, pos_tzero + normal_component*normal(t=t0), rgbcolor=(1,0,1))
#
# accumulate the graphic based on checkboxes
#
picture = path
if pos_check:
picture = picture + pos
if vel_check:
picture = picture + vel
if tan_check:
picture = picture+ tan
if nor_check:
picture = picture + nor
if bin_check:
picture = picture + bin
if acc_check:
picture = picture + acc
if tancomp_check:
picture = picture + tancomp
if norcomp_check:
picture = picture + norcomp
#
# print textual info
#
print("Position vector: r(t)=", position)
print("Speed is ", N(speed(t=t0)))
print("Curvature is ", N(curvature))
## print("Torsion is ", N(torsion))
print()
print("Right-click on graphic to zoom to 400%")
print("Drag graphic to rotate")
#
# show accumulated graphical info
#
show(picture, aspect_ratio=[1,1,1])
```

## Options

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Date: 23 Jul 2020 23:58

Submitted by: Zane Corbiere