Sample PlutoJL worksheet for basics of 2D vectors

vectors_ws1.jl

### A Pluto.jl notebook ###
# v0.19.42

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ 86b7bdf0-ff83-49b3-b1bf-898958ae0566
using PlutoUI, Plots, HypertextLiteral

# ╔═╡ 8658a023-30c2-4ca8-8cb5-706e945cd5e1
begin
	
function vector(x...)
    [x...]
end

"""
Calculates the 'inner' or 'dot' or 'scalar' product
of the two given vectors. Those are all names that
refer to the same operation. This product will distribute
over vector addition just like ordinary multiplication
of numbers.
"""
function inner_product(va, vb)
    va' * vb
end

"""
Interprets a 2D vector as a complex number with the
x axis as the "real" part and the y axis as the 
"imaginary" part.
"""
function vector2complex(v :: AbstractVector{T}) where {T <: Real}
    @assert length(v) == 2 "Vector must be 2 dimensional to conver to a complex number"
    v[1] + im * v[2]
end

"""
Interprets a complex number as a 2D vector with the
x axis as the "real" part and the y axis as the
"imaginary" part.
"""
function complex2vector(c :: ComplexF64)
    [real(c), imag(c)]
end

function measure(v :: AbstractVector{T}) where {T <: Real}
    sqrt(inner_product(v, v))
end

function unit_vector(v :: AbstractVector{T}) where {T <: Real}
    v ./ measure(v)
end

function graph(vecs :: AbstractVector{Pair{Vector{T},Vector{T}}}; s=6) where {T <: Real}
	v1 = vecs[1]
	w = 250
	step = ceil(s/5)
	axcolor = :black
	kv = (; aspect_ratio=1.0, arrow=true, xlims=(-s,s), ylims=(-s,s),
			xticks=-s:step:s, yticks=-s:step:s, size=(w,w), framestyle=:origin,
			draw_arrow=true, x_draw_arrow=true, x_foreground_color_axis=axcolor, x_foreground_color_border=axcolor, y_foreground_color_axis=axcolor, y_foreground_color_border=axcolor, lw=2, legend=false)
	p = plot([v1.first[1], v1.second[1]], [v1.first[2], v1.second[2]]; kv...)
	for i in 2:length(vecs)
		v = vecs[i]
		plot!(p, [v.first[1], v.second[1]], [v.first[2], v.second[2]]; kv...)
	end
	return p		
end

function graph(vec :: Pair{Vector{T}, Vector{T}};s=6) where {T <: Real}
	graph([vec];s)
end
	
function graph(vecs :: AbstractVector{Vector{T}};s=6) where {T <: Real}
	origin = [0,0]
	graph([origin => v for v in vecs];s)
end

function graph(comps :: AbstractVector{Pair{ComplexF64,ComplexF64}};s=6)
	graph([complex2vector(c.first) => complex2vector(c.second) for c in comps];s)
end

function graph(comps :: AbstractVector{ComplexF64}; s=6)
	graph(complex2vector.(comps);s)
end

function graphseq(vecs :: AbstractVector{Vector{T}};s=6) where {T <: Real}
	graph([vecs[i-1] => vecs[i] for i in 2:length(vecs)];s)
end	

function randvector(s)
	v = [0,0]
	while v == [0,0]
		v = vector(rand(-s:s), rand(-s:s))
	end
	return v
end
	
function showcomplex(c::Complex{T}) where {T <: Real}
	md"($(real(c)) + $(imag(c))``i``)"
end

function dec2(n::Float64)
	round(Int,n*100)/100
end
	
	nothing
end

# ╔═╡ 6a160233-b487-4e54-8a3a-161349eba1f0
md"""
# Vectors

Below, you'll represent two-dimensions vectors as a pair of numbers giving the ``(x,y)`` coordinates of a point on a plane. A vector will be written as `vector(x,y)`.
"""

# ╔═╡ 232ed6f8-fc30-476e-ac68-0beb5145addf
md"""
$(@bind vadd Button("Once more"))
"""

# ╔═╡ aaafca26-63e3-4129-8405-0dcf7d260394
begin
	vadd
	vadd_blue = randvector(3)
	vadd_red = randvector(3)
	md"""
	## Addition
	
	You're given two vector ``\vec{A}`` (colour blue) and ``\vec{B}`` (colour red). Your task is to compute the coordinates corresponding to vector ``\vec{C} = \vec{A} + \vec{B}`` (colour green).
	"""
end


# ╔═╡ 96616549-f7c1-4b57-a81f-f79aba276a1d
	md"""
	- ``\vec{A}`` (blue vector) is ($(vadd_blue[1]), $(vadd_blue[2]))
	- ``\vec{B}`` (red vector) is ($(vadd_red[1]), $(vadd_red[2]))
	- ``\vec{C} (\text{green vector}) =`` ($(@bind vadd_cx NumberField(-10:10, default=1)), $(@bind vadd_cy NumberField(-10:10, default=1)))
	"""

# ╔═╡ dbac50d9-867d-4584-a918-7af0da3f665b
md"""
**Evaluation**: $(if vadd_cx == vadd_blue[1] + vadd_red[1] "✓ x coordinate" else "😕 incorrect x coordinate" end), $(if vadd_cy == vadd_blue[2] + vadd_red[2] "✓ y coordinate" else "😕 incorrect y coordinate" end)
"""

# ╔═╡ c5d41955-b407-42e6-ab6b-66b3171e6374
begin
	vadd_c = vector(vadd_cx, vadd_cy)
	o = vector(0,0)
	graph([
		o => vadd_blue,
		vadd_blue => (vadd_blue + vadd_red),
		o => vadd_c
	]; s=6)
end

# ╔═╡ 04c37b79-7ecd-45ab-98c3-53a156551239
md"""
$(@bind vneg Button("Once more"))
"""

# ╔═╡ b551f901-56dc-4176-b853-ee4dfa79438d
begin
	vneg
	vneg_a = randvector(3)
md"""
## Flipping a vector by 180°

You're given a vector ``\vec{A}`` (colour blue). Your task is to compute the coordinates corresponding to vector ``\vec{B}`` that's of the same length as ``\vec{A}`` but points in the opposite direction..
"""
end


# ╔═╡ f2efcc08-d04c-4dc3-9546-6a07c9b93eda
	md"""
	- ``\vec{A}``(blue vector) is ($(vneg_a[1]), $(vneg_a[2]))
	- ``\vec{B} = (b_x,b_y) =`` ($(@bind vneg_bx NumberField(-10:10, default=1)), $(@bind vneg_by NumberField(-10:10, default=1)))
	"""

# ╔═╡ cf65c50a-0cc4-4054-9af1-d3acec4dc01f
md"""
**Evaluation**: $(if vneg_bx + vneg_a[1] == 0 "✓ x coordinate" else "😕 incorrect x coordinate" end), $(if vneg_by + vneg_a[2] == 0 "✓ y coordinate" else "😕 incorrect y coordinate" end)
"""

# ╔═╡ e7d73a50-625a-4c96-b310-14ea73c974f3
begin
	vneg_b = vector(vneg_bx, vneg_by)
	graph([
		o => vneg_a,
		o => vneg_b
	]; s=3)
end

# ╔═╡ 54af3aaa-f6be-46bf-a5ea-eef83a8fd7c1
md"""
$(@bind vsub Button("Once more"))
"""

# ╔═╡ 9288b8a3-fd12-4fba-a1ed-163023d56db7
begin
	vsub
	vsub_blue = randvector(3)
	vsub_red = randvector(3)
md"""
## Subtraction

You're given two vector ``\vec{A}`` (colour blue) and ``\vec{B}`` (colour red). Your task is to compute the coordinates corresponding to vector ``\vec{C}(\text{green}) = \vec{A}(\text{blue}) - \vec{B}(\text{red})``.
"""
end


# ╔═╡ 561da377-d2fe-4225-80fb-d8b547fd1f3b
	md"""
	- ``\vec{A}``(blue vector) is ($(vsub_blue[1]), $(vsub_blue[2]))
	- ``\vec{B}``(red vector) is ($(vsub_red[1]), $(vsub_red[2]))
	- ``\vec{C} (\text{green vector}) = (c_x, c_y) =`` ($(@bind vsub_cx NumberField(-10:10, default=1)), $(@bind vsub_cy NumberField(-10:10, default=1)))
	- The purple vector is the red vector flipped by 180°.
	"""

# ╔═╡ 0fbc8d0d-e875-4928-a4f6-9b9f7bc68bce
md"""
**Evaluation**: $(if vsub_cx == vsub_blue[1] - vsub_red[1] "✓ x coordinate" else "😕 incorrect x coordinate" end), $(if vsub_cy == vsub_blue[2] - vsub_red[2] "✓ y coordinate" else "😕 incorrect y coordinate" end)
"""

# ╔═╡ 5cfebd64-a65f-4c4c-9ff3-d67d6de2d36a
begin
	vsub_c = vector(vsub_cx, vsub_cy)
	graph([
		o => vsub_blue,
		o => vsub_red,
		o => vsub_c,
		vsub_blue => (vsub_blue-vsub_red)
	]; s=6)
end

# ╔═╡ e4f9720a-03d8-4a51-8f7a-c91f4415a4c6
@bind randblue1 Button("Once more")

# ╔═╡ d94536f4-cc9c-431f-90a7-08b20a682c4f
begin
	bluevec = let
		randblue1
		randvector(3)
	end
md"""
## Anti-clockwise rotation by 90°
Change the coordinates of the `red` vector below so that it is **perpendicular** to the `blue` vector and is rotated with respect to the blue vector in the **anti-clockwise** direction. The two vectors must also be of the **same length**.
"""
end


# ╔═╡ 685c09b5-d09c-40be-ba20-11bafe907230
md"""
- blue vector = ($(bluevec[1]), $(bluevec[2]))
- red vector = ``(x,y)`` = ($(@bind redx NumberField(-10:10, default=1)), $(@bind redy NumberField(-10:10, default=1)))
"""

# ╔═╡ 506405ee-6144-438b-a5b1-4de23dc7de74
begin
	# TASK: Change the x and y values of the "red" vector so that it
	# is perpendicular to the blue vector and is rotated w.r.t. the blue
	# vector in the anti-clockwise direction. The two vectors must also be
	# of the same length.
	red = vector(redx, redy)
	graph([bluevec, red];s=3)
end

# ╔═╡ c7b40905-d0bd-462e-b3c0-736b884d616a
md"""
**Evaluation**: $(if red' * bluevec == 0 "✓ angle" else "😕 NOT at right angles" end), $(if isapprox(red' * red, bluevec' * bluevec) "✓ length" else "😕 NOT of same length" end), $(if bluevec[1] * red[2] - bluevec[2] * red[1] > 0 "✓ rotation direction" else "😕 MUST be anti-clockwise rotation" end)

"""

# ╔═╡ 6af1ff5c-bdf1-41c6-a82c-c16b16125905
@bind vclock Button("Once more")

# ╔═╡ a0ba2b3b-7089-4f95-be96-5ede2fb2223e
begin
	vclock
	vclock_a = randvector(3)
md"""
## Clockwise rotation by 90°
Change the coordinates of the `red` vector below so that it is **perpendicular** to the `blue` vector and is rotated with respect to the blue vector in the **clockwise** direction. The two vectors must also be of the **same length**.
"""
end


# ╔═╡ e32b8cc1-18f2-4186-b6bb-24c4258fc287
md"""
- blue vector = ($(vclock_a[1]), $(vclock_a[2]))
- red vector = ``(x,y)`` = ($(@bind vclock_bx NumberField(-10:10, default=1)), $(@bind vclock_by NumberField(-10:10, default=1)))
"""

# ╔═╡ 90f80e0a-24c3-4067-80ff-928d548339d6
begin
	vclock_b = vector(vclock_bx, vclock_by)
md"""
**Evaluation**: $(if vclock_b' * vclock_a == 0 "✓ angle" else "😕 NOT at right angles" end), $(if isapprox(vclock_b' * vclock_b, vclock_a' * vclock_a) "✓ length" else "😕 NOT of same length" end), $(if vclock_a[1] * vclock_b[2] - vclock_a[2] * vclock_b[1] < 0 "✓ rotation direction" else "😕 MUST be clockwise rotation" end)

"""
end

# ╔═╡ 7f0e233d-0ceb-4e67-a790-6ac367978977
begin
	# TASK: Change the x and y values of the "red" vector so that it
	# is perpendicular to the blue vector and is rotated w.r.t. the blue
	# vector in the anti-clockwise direction. The two vectors must also be
	# of the same length.
	graph([vclock_a, vclock_b];s=3)
end

# ╔═╡ 552cb938-fbc1-4964-92f0-0164fdb377ad
@bind vscale Button("Once more")

# ╔═╡ 5c3e3ac5-2de0-4b7c-bf40-b151db49a25b
begin
	vscale
	vscale_a = Float64.(vector(rand(2:5),rand(2:5)) .* vector(rand(-1:2:1), rand(-1:2:1)))
	md"""
	# Scaling a vector
	Multiplying a vector by a scalar (i.e. a number) "scales" the vector. In other words, it increases or decreases its length without changing its direction if the number is a positive real number, and flips its direction if it is negative real number.
	"""
end

# ╔═╡ a5bfc444-bf98-4c29-bc4b-e47064c8fed4
@bind vscale_factor Slider(-3:0.1:3)

# ╔═╡ 017e4dc4-a8b1-43f8-8638-6289f5adbfb4
md"""
- ``\vec{A} = `` ($(vscale_a[1]), $(vscale_a[2]))
- Scale factor = ``\alpha`` = $(vscale_factor)
- ``\alpha \vec{A} = `` ($(dec2(vscale_factor * vscale_a[1])), $(dec2(vscale_factor * vscale_a[2])))
"""

# ╔═╡ 688a7f50-f192-427d-80b8-94a3ea9f468d
begin
	fo = Float64.(o)
	graph([
		fo => vscale_a,
		fo => vscale_factor * vscale_a
	], s=15)
end

# ╔═╡ f3827bb0-f4e7-41d7-bf80-06b1c1684de6
@bind vdot Button("Once more")

# ╔═╡ 7060fb3c-b601-4ac9-aa29-dae4058c2138
begin
	vdot
	vdot_a = randvector(5)
	vdot_b = randvector(5)
md"""
## Inner product of two vectors
**Info**: The "inner product" of two vectors computes a plain number, also known as a "**scalar**". The "inner product" of two vectors is also known as the "dot product" or the "scalar product". It is usually written ``\vec{A}\boldsymbol{\cdot}\vec{B}``, but in some other contexts, it may be written as ``\langle\vec{A},\vec{B}\rangle`` or ``\braket{A|B}``. Know that they all mean the same concept.

If you represent a vector as a column matrix with one column like this - 
``\begin{bmatrix} x\\y \end{bmatrix}``, then the scalar product of two vectors can be represented as the matrix product of the transpose of one vector with the other vector - like this - ``\begin{bmatrix}a_x \\ a_y\end{bmatrix}^T\begin{bmatrix}b_x \\ b_y\end{bmatrix} = \begin{bmatrix}a_x & a_y\end{bmatrix}\begin{bmatrix}b_x \\ b_y\end{bmatrix} = a_x b_x + a_y b_y``. Recall that the "transpose" of a column vector gives a row vector - i.e. the transpose of ``\begin{bmatrix}a_x \\ a_y\end{bmatrix}``, written as ``\begin{bmatrix}a_x \\ a_y\end{bmatrix}^T = \begin{bmatrix}a_x & a_y\end{bmatrix}``.

**Vector length:**
- The length of a vector ``\vec{A}`` is written as ``|\vec{A}|`` and is defined as ``|\vec{A}|^2 = \vec{A} \cdot \vec{A}``.

**Distributive law:**
- ``\vec{A} \boldsymbol{\cdot} (\vec{B} + \vec{C}) = \vec{A} \boldsymbol{\cdot} \vec{B} + \vec{A} \boldsymbol{\cdot} \vec{C}``
- ``(\vec{A} + \vec{B}) \boldsymbol{\cdot} \vec{C} = \vec{A} \boldsymbol{\cdot} \vec{C} + \vec{B} \boldsymbol{\cdot} \vec{C}``

**Scaling law (also known as "linearity"):**
- ``\vec{A} \boldsymbol{\cdot} (\alpha \vec{B}) = \alpha (\vec{A} \boldsymbol{\cdot} \vec{B})``
- ``(\alpha \vec{A}) \boldsymbol{\cdot} \vec{B} = \alpha (\vec{A} \boldsymbol{\cdot} \vec{B})``

**Task:** Calculate the inner product of the two given vectors ``\vec{A}`` and ``\vec{B}``

"""
end

# ╔═╡ 06107497-0e35-47ec-ac28-0aaa9c4291d5
md"""
``\vec{A}`` = ($(vdot_a[1]), $(vdot_a[2]))

``\vec{B}`` = ($(vdot_b[1]), $(vdot_b[2]))

``\vec{A} \cdot \vec{B} = `` $(@bind vdot_val NumberField(-100:100, default=0))
"""

# ╔═╡ ed53263b-d934-4871-9e2a-33d3c00ea6eb
md"""
**Evaluation**: $(if vdot_val == inner_product(vdot_a,vdot_b) "✓ correct" else "😕 incorrect" end)
"""

# ╔═╡ fcfdca57-e61f-40a4-b6a3-ab3bb846b3a7
md"""
# Quiz

(TODO)
"""

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[[deps.FFMPEG]]
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[[deps.FFMPEG_jll]]
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[[deps.OpenSSL]]
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[[deps.Parsers]]
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git-tree-sha1 = "330f955bc41bb8f5270a369c473fc4a5a4e4d3cb"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.6+0"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "691634e5453ad362044e2ad653e79f3ee3bb98c3"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.39.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e92a1a012a10506618f10b7047e478403a046c77"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.5.0+0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+1"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e678132f07ddb5bfa46857f0d7620fb9be675d3b"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.6+0"

[[deps.eudev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "gperf_jll"]
git-tree-sha1 = "431b678a28ebb559d224c0b6b6d01afce87c51ba"
uuid = "35ca27e7-8b34-5b7f-bca9-bdc33f59eb06"
version = "3.2.9+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "a68c9655fbe6dfcab3d972808f1aafec151ce3f8"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.43.0+0"

[[deps.gperf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3516a5630f741c9eecb3720b1ec9d8edc3ecc033"
uuid = "1a1c6b14-54f6-533d-8383-74cd7377aa70"
version = "3.1.1+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "1827acba325fdcdf1d2647fc8d5301dd9ba43a9d"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.9.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.8.0+1"

[[deps.libevdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "141fe65dc3efabb0b1d5ba74e91f6ad26f84cc22"
uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
version = "1.11.0+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libinput_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "eudev_jll", "libevdev_jll", "mtdev_jll"]
git-tree-sha1 = "ad50e5b90f222cfe78aa3d5183a20a12de1322ce"
uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
version = "1.18.0+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "d7015d2e18a5fd9a4f47de711837e980519781a4"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.43+1"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.mtdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "814e154bdb7be91d78b6802843f76b6ece642f11"
uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
version = "1.1.6+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.52.0+1"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+1"
"""

# ╔═╡ Cell order:
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# ╟─fcfdca57-e61f-40a4-b6a3-ab3bb846b3a7
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002