problem02

2. 3D. Get good at vectors.

Assume that the positions relative to an origin of four random points, which are randomly located in space are given as \(\vec{\mathbf{r}}_A\), \(\vec{\mathbf{r}}_B\), \(\vec{\mathbf{r}}_C\) and \(\vec{\mathbf{r}}_D\). Assume force \(\vec{\mathbf{F}}\) is given. For each problem below write a single vector formula (one for each problem) that answers the question.

1. The points A and B define an infinite line. So do the points C and D. Find the distance between these two lines. ‘The’ distance means ‘the minimum distance’, that is the length of the shortest line segment connecting the two lines. Either write a formula (or sequence of formulas), or write computer code that gives the answer, or both.
   
2. Same problem as above, but also find the end points of the shortest line segment.
   
3. Find the volume of the tetrahedron ABCD (you should reason-out and not quote any formulas for the volume of a tetrahedron, that is, see if you can derive the formula: ’volume = one third base times height’).
   
4. Assume points A, B and C are fixed to a structure. All three are connected, by massless rods, to a ball and socket at each end, to point D. At point D the force F is applied. Find the tension in bar AD. Find a formula for the answer, or write computer code to find the answer, or both. The goal is to find a formula for the tension in terms of the positions and the force vector.

1 Write formula or code for the length of the shortest line segment

The lines can be represented as:

\[l_1: \vec{p}_1(\lambda_1) = \quad \vec{\mathbf{r}}_A + \lambda_1 (\vec{\mathbf{r}}_B-\vec{\mathbf{r}}_A)\]

\[l_2: \vec{p}_2(\lambda_2) = \quad \vec{\mathbf{r}}_C + \lambda_2 (\vec{\mathbf{r}}_D-\vec{\mathbf{r}}_C)\]

Hence, for two points on either lines, \(p_1\), \(p_2\), the distance is:

\[D(\lambda_1, \lambda_2) = \quad \left|\left| \mathbf{\vec{p}}_2 - \mathbf{\vec{p}}_1 \right|\right|_n\]

Which can be framed as an optimisation problem, minimum length, \(\hat{s}\):

\[\hat{s} = \quad \min_{\lambda_1, \lambda_2} || \left( \vec{\mathbf{r}}_A + \lambda_1 (\vec{\mathbf{r}}_B-\vec{\mathbf{r}}_A) - (\vec{\mathbf{r}}_C + \lambda_2 (\vec{\mathbf{r}}_D-\vec{\mathbf{r}}_C)) \right) ||_n\]

or,

\[\hat{s} \leftarrow D\left\{ \nabla D(\lambda_1, \lambda_2) = 0 \right\}\]

In file ./MinimumDistanceBetweenTwoLines/src/MinimumDistanceBetweenTwoLines.jl:

module MinimumDistanceBetweenTwoLines

import Symbolics
import LinearAlgebra

# problem setup
const N = 3
Symbolics.@variables r⃗ₛ[1:N] r⃗ₜ[1:N] r⃗ᵤ[1:N] r⃗ᵥ[1:N]
Symbolics.@variables λ₁ λ₂

p⃗₁ = r⃗ₛ + λ₁ * (r⃗ₜ - r⃗ₛ)
p⃗₂ = r⃗ᵤ + λ₂ * (r⃗ᵥ - r⃗ᵤ)
s⃗ = p⃗₂ - p⃗₁
D² = LinearAlgebra.dot(s⃗, s⃗)
D² = Symbolics.scalarize(D²)
∇D² = Symbolics.gradient(D², [λ₁, λ₂])
eq = ∇D² .~ 0

lambdas_symbolic = Dict([λ₁, λ₂] .=> Symbolics.symbolic_linear_solve(eq, [λ₁, λ₂]))
p̂₁_symbolic, p̂₂_symbolic = Symbolics.substitute.([p⃗₁, p⃗₂], Ref(lambdas_symbolic))
ŝ_symbolic = Symbolics.substitute.(D², Ref(lambdas_symbolic))

#...

Note that in the code, we are optimising for distance squared, and the \(s\) symbols in the code represents the distance squared.

In the same file, we take specific values:

#...

# evaluation
subs = Dict(
    r⃗ₛ => [0.0, 0.0, 0.0],
    r⃗ₜ => [1.0, 0.0, 0.0],
    r⃗ᵤ => [0.0, 0.0, 1.0],
    r⃗ᵥ => [0.0, 1.0, 2.0],
    # λ₁ => 3,
    # λ₂ => 4,
)
∇D²_eval = Symbolics.substitute.(∇D², Ref(subs))

eq_eval = Symbolics.substitute.(eq, Ref(subs))

result_eval = Dict(symbol => Symbolics.substitute(expresion, subs) for (symbol, expresion) in lambdas_symbolic)
merged_eval = merge(subs, result_eval)

ŝ_eval = Symbolics.substitute.(D², Ref(merged_eval))
p̂₁_eval, p̂₂_eval = Symbolics.substitute.([p⃗₁, p⃗₂], Ref(merged_eval))

println("=== Substituted results: =====")
println("Λ̂  : ", result_eval)
println("ŝ  = ", ŝ_eval)
println("p̂₁ = ", p̂₁_eval)
println("p̂₂ = ", p̂₂_eval)

println("\n=== Symbolic results: =====")
println("Λ̂  : ", lambdas_symbolic)
println("ŝ  = ", Symbolics.scalarize(ŝ_symbolic))
println("p̂₁ = ", Symbolics.scalarize(p̂₁_symbolic))
println("p̂̂₂ = ", Symbolics.scalarize(p̂₂_symbolic))

end # module MinimumDistanceBetweenTwoLines

This produces the following:

julia> include("src/MinimumDistanceBetweenTwoLines.jl")
=== Substituted results: =====
Λ̂  : Dict{Symbolics.Num, Float64}(λ₂ => -0.5, λ₁ => -0.0)
ŝ  = 0.5
p̂₁ = [0.0, 0.0, 0.0][Base.OneTo(3)]
p̂₂ = [0.0, -0.5, 0.5][Base.OneTo(3)]

=== Symbolic results: =====
Λ̂  : Dict{Symbolics.Num, SymbolicUtils.BasicSymbolic{Real}}(λ₂ => ((-(2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) + 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(⃗rₛ[3] - r⃗ₜ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) + 2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ᵤ[1] - r⃗ₛ[1]) + 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ᵤ[2] - r⃗ₛ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(-r⃗ᵤ[3] + r⃗ᵥ[3])) / ((-((-2(-⃗rᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))^2)) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2((-r⃗ᵤ[1] + r⃗ᵥ[1])^2) - 2((-r⃗ᵤ[2] + r⃗ᵥ[2])^2) - 2((-r⃗ᵤ[3] + r⃗ᵥ[3])^2)), λ₁ => ((-((-(2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) + 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) + 2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ᵤ[1] - r⃗ₛ[1]) + 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ᵤ[2] - r⃗ₛ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(-r⃗ᵤ[3] + r⃗ᵥ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / ((-((-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))^2)) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2((-r⃗ᵤ[1] + r⃗ᵥ[1])^2) - 2((-r⃗ᵤ[2] + r⃗ᵥ[2])^2) - 2((-r⃗ᵤ[3] + r⃗ᵥ[3])^2)) + 2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) + 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3])) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)))
ŝ  = (r⃗ᵤ[1] - r⃗ₛ[1] + ((-r⃗ᵤ[1] + r⃗ᵥ[1])*(((-2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) + 2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ᵤ[1] - r⃗ₛ[1]) + 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ᵤ[2] - r⃗ₛ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(-r⃗ᵤ[3] + r⃗ᵥ[3]))) / ((-((-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))^2)) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2((-r⃗ᵤ[1] + r⃗ᵥ[1])^2) - 2((-r⃗ᵤ[2] + r⃗ᵥ[2])^2) - 2((-r⃗ᵤ[3] + r⃗ᵥ[3])^2)) + (((((-(-2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(r⃗ᵤ[3] - ⃗rₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ᵤ[1] - r⃗ₛ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ᵤ[2] - r⃗ₛ[2]) - 2(r⃗ᵤ[3] - r⃗ₛ[3])*(-r⃗ᵤ[3] + r⃗ᵥ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / ((-((-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))^2)) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2((-r⃗ᵤ[1] + r⃗ᵥ[1])^2) - 2((-r⃗ᵤ[2] + r⃗ᵥ[2])^2) - 2((-r⃗ᵤ[3] + r⃗ᵥ[3])^2)) + 2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) + 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(r⃗ₛ[1] - r⃗ₜ[1])) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)))^2 + (r⃗ᵤ[2] - r⃗ₛ[2] + (((((-(-2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ᵤ[1] - r⃗ₛ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ᵤ[2] - r⃗ₛ[2]) - 2(r⃗ᵤ[3] - r⃗ₛ[3])*(-r⃗ᵤ[3] + r⃗ᵥ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / ((-((-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))^2)) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2((-r⃗ᵤ[1] + r⃗ᵥ[1])^2) - 2((-r⃗ᵤ[2] + r⃗ᵥ[2])^2) - 2((-r⃗ᵤ[3] + r⃗ᵥ[3])^2)) + 2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) + 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(r⃗ₛ[2] - r⃗ₜ[2])) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) + ((((-2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) + 2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ᵤ[1] - r⃗ₛ[1]) + 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ᵤ[2] - r⃗ₛ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(-r⃗ᵤ[3] + r⃗ᵥ[3]))*(-r⃗ᵤ[2] + r⃗ᵥ[2])) / ((-((-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))^2)) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2((-r⃗ᵤ[1] + r⃗ᵥ[1])^2) - 2((-r⃗ᵤ[2] + r⃗ᵥ[2])^2) - 2((-r⃗ᵤ[3] + r⃗ᵥ[3])^2)))^2 + (r⃗ᵤ[3] - r⃗ₛ[3] + ((((((-2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) + 2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ᵤ[1] - r⃗ₛ[1]) + 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ᵤ[2] - r⃗ₛ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(-r⃗ᵤ[3] + r⃗ᵥ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / ((-((-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))^2)) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2((-r⃗ᵤ[1] + r⃗ᵥ[1])^2) - 2((-r⃗ᵤ[2] + r⃗ᵥ[2])^2) - 2((-r⃗ᵤ[3] + r⃗ᵥ[3])^2)) - 2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(-r⃗ₛ[3] + r⃗ₜ[3])) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) + ((((-2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) + 2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ᵤ[1] - r⃗ₛ[1]) + 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ᵤ[2] - r⃗ₛ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(-r⃗ᵤ[3] + r⃗ᵥ[3]))*(-r⃗ᵤ[3] + r⃗ᵥ[3])) / ((-((-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))^2)) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2((-r⃗ᵤ[1] + r⃗ᵥ[1])^2) - 2((-r⃗ᵤ[2] + r⃗ᵥ[2])^2) - 2((-r⃗ᵤ[3] + r⃗ᵥ[3])^2)))^2
p̂₁ = Symbolics.Num[r⃗ₛ[1] + ((((((2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) + 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(⃗rₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ᵤ[1] - r⃗ₛ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ᵤ[2] - r⃗ₛ[2]) - 2(r⃗ᵤ[3] - r⃗ₛ[3])*(-r⃗ᵤ[3] + r⃗ᵥ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / ((-((-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))^2)) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2((-r⃗ᵤ[1] + r⃗ᵥ[1])^2) - 2((-r⃗ᵤ[2] + r⃗ᵥ[2])^2) - 2((-r⃗ᵤ[3] + r⃗ᵥ[3])^2)) + 2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) + 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(-r⃗ₛ[1] + r⃗ₜ[1])) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)), r⃗ₛ[2] + ((((((2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) + 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ᵤ[1] - r⃗ₛ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ᵤ[2] - r⃗ₛ[2]) - 2(r⃗ᵤ[3] - r⃗ₛ[3])*(-r⃗ᵤ[3] + r⃗ᵥ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / ((-((-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - ⃗rₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))^2)) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2((-r⃗ᵤ[1] + r⃗ᵥ[1])^2) - 2((-r⃗ᵤ[2] + r⃗ᵥ[2])^2) - 2((-r⃗ᵤ[3] + r⃗ᵥ[3])^2)) + 2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) + 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(-r⃗ₛ[2] + r⃗ₜ[2])) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)), r⃗ₛ[3] + ((((((2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) + 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ᵤ[1] - r⃗ₛ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ᵤ[2] - r⃗ₛ[2]) - 2(r⃗ᵤ[3] - r⃗ₛ[3])*(-r⃗ᵤ[3] + r⃗ᵥ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / ((-((-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))^2)) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2((-r⃗ᵤ[1] + r⃗ᵥ[1])^2) - 2((-r⃗ᵤ[2] + r⃗ᵥ[2])^2) - 2((-r⃗ᵤ[3] + r⃗ᵥ[3])^2)) + 2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) + 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(-r⃗ₛ[3] + r⃗ₜ[3])) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2))]
p̂̂₂ = Symbolics.Num[r⃗ᵤ[1] + ((-r⃗ᵤ[1] + r⃗ᵥ[1])*(((-2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(-2(-⃗rᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) + 2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ᵤ[1] - r⃗ₛ[1]) + 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ᵤ[2] - r⃗ₛ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(-r⃗ᵤ[3] + r⃗ᵥ[3]))) / ((-((-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))^2)) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2((-r⃗ᵤ[1] + r⃗ᵥ[1])^2) - 2((-r⃗ᵤ[2] + r⃗ᵥ[2])^2) - 2((-r⃗ᵤ[3] + r⃗ᵥ[3])^2)), r⃗ᵤ[2] + ((((-2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) + 2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ᵤ[1] - r⃗ₛ[1]) + 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ᵤ[2] - r⃗ₛ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(-r⃗ᵤ[3] + r⃗ᵥ[3]))*(-r⃗ᵤ[2] + r⃗ᵥ[2])) / ((-((-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))^2)) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2((-r⃗ᵤ[1] + r⃗ᵥ[1])^2) - 2((-r⃗ᵤ[2] + r⃗ᵥ[2])^2) - 2((-r⃗ᵤ[3] + r⃗ᵥ[3])^2)), r⃗ᵤ[3] + ((((-2(r⃗ᵤ[1] - r⃗ₛ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(r⃗ᵤ[2] - r⃗ₛ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(r⃗ᵤ[3] - r⃗ₛ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))*(-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) + 2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ᵤ[1] - r⃗ₛ[1]) + 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ᵤ[2] - r⃗ₛ[2]) + 2(r⃗ᵤ[3] - r⃗ₛ[3])*(-r⃗ᵤ[3] + r⃗ᵥ[3]))*(-r⃗ᵤ[3] + r⃗ᵥ[3])) / ((-((-2(-r⃗ᵤ[1] + r⃗ᵥ[1])*(r⃗ₛ[1] - r⃗ₜ[1]) - 2(-r⃗ᵤ[2] + r⃗ᵥ[2])*(r⃗ₛ[2] - r⃗ₜ[2]) - 2(-r⃗ᵤ[3] + r⃗ᵥ[3])*(r⃗ₛ[3] - r⃗ₜ[3]))^2)) / (-2((r⃗ₛ[1] - r⃗ₜ[1])^2) - 2((r⃗ₛ[2] - r⃗ₜ[2])^2) - 2((r⃗ₛ[3] - r⃗ₜ[3])^2)) - 2((-r⃗ᵤ[1] + r⃗ᵥ[1])^2) - 2((-r⃗ᵤ[2] + r⃗ᵥ[2])^2) - 2((-r⃗ᵤ[3] + r⃗ᵥ[3])^2))]
Main.MinimumDistanceBetweenTwoLines

2 Also find the end points of the shortest line segment

The formulation above can be reused, the optimal lambdas \(\hat{\Lambda} = (\lambda_1, \lambda_2)\), correspond to the closest points through the equation of the lines.

\[\left(\hat{\lambda}_1, \hat{\lambda}_2\right) = \underset{\lambda_1, \lambda_2}{\operatorname*{argmin}} \left\{ D(\lambda_1, \lambda_2) \right\}\]

A specific example’s numerical output was given in the previous section.

3 Find the volume of the tetrahedron

For any 2d shape extended to a point, the volume is \(\frac{1}{3} \text{Base-Area} \times \text{height}\). Why? Take a slice, the planar dimensions reduce linearly to zero, so the area reduces quadratically. Integrate the function for the area as a function of the height from base to the tip.

This way, we can see that a tetrahedron is geometric one-sixth of a parallelapiped, therefore, volume \(V(\vec{\mathbf{r}}_a, \vec{\mathbf{r}}_b, \vec{\mathbf{r}}_c, \vec{\mathbf{r}}_d)\) is (assuming the vectors are in three dimensions for cross product to work):

\[V(\vec{\mathbf{r}}_a, \vec{\mathbf{r}}_b, \vec{\mathbf{r}}_c, \vec{\mathbf{r}}_d) = \frac{1}{6} \left|(\vec{\mathbf{r}}_c - \vec{\mathbf{r}}_b) \times (\vec{\mathbf{r}}_c - \vec{\mathbf{r}}_a) \cdot (\vec{\mathbf{r}}_d - \vec{\mathbf{r}}_d)\right|\]

4 Write formula or code for the tensions in the rods

\[\vec{\mathbf{F}} + T_{ad}\hat{r}_{ad} + T_{bd}\hat{r}_{bd} + T_{cd}\hat{r}_{cd} = \vec{\mathbf{0}}\]

There was an in-class mention about Cramer’s rule to solve this. There are three equations and three unknowns, but one could use Gaussian-elimination.

The other way to solve this using vector algebra is to dot the whole equation with a vector which is perpendicular to two of the three tension vectors. For example, dot the equation by \(\hat{r}_{cd} \times \hat{r}_{bd}\).