use quarto, not Pluto to render pages

CwJ/differentiable_vector_calculus/vectors.jmd

@@ -363,9 +363,9 @@ Starting with three vectors, we can create three orthogonal vectors using projec
 Let's begin with three vectors in $R^3$:
 
 ```julia;
-u⃗ = [1, 2, 3]
-v⃗ = [1, 1, 2]
-w⃗ = [1, 2, 4]
+u = [1, 2, 3]
+v = [1, 1, 2]
+w = [1, 2, 4]
 ```
 
 We can find a vector from `v` orthogonal to `u` using:
@@ -374,14 +374,14 @@ We can find a vector from `v` orthogonal to `u` using:
 unit_vec(u) = u / norm(u)
 projection(u, v) = (u ⋅ unit_vec(v)) * unit_vec(v)
 
-v⃗⟂ = v⃗ - projection(v⃗, u⃗)
-w⃗⟂ = w⃗ - projection(w⃗, u⃗) - projection(w⃗, v⃗⟂)
+vₚ = v - projection(v, u)
+wₚ = w - projection(w, u) - projection(w, vₚ)
 ```
 
 We can verify the orthogonality through:
 
 ```julia;
-u⃗ ⋅ v⃗⟂, u⃗ ⋅ w⃗⟂, v⃗⟂ ⋅ w⃗⟂
+u ⋅ vₚ, u ⋅ wₚ, vₚ ⋅ wₚ
 ```
 
 This only works when the three vectors do not all lie in the same plane. In general, this is the beginning of the [Gram-Schmidt](https://en.wikipedia.org/wiki/Gram-Schmidt_process) process for creating *orthogonal* vectors from a collection of vectors.
@@ -839,11 +839,8 @@ The volume of a parallelepiped is the area of a base parallelogram times the hei
 that is, the area of the parallelepiped. Wait, what about $(\vec{v}\times\vec{u})\cdot\vec{w}$? That will have an opposite sign. Yes, in the above, there is an assumption that $\vec{n}$ and $\vec{w}$ have a an angle between them within $[0, \pi/2]$, otherwise an absolute value must be used, as volume is non-negative.
 
 
-```julia; echo=false
-note(L"""
-The triple-scalar product, $\vec{u}\cdot(\vec{v}\times\vec{w})$, gives the volume of the parallelepiped up to sign. If the sign of this is positive, the ``3`` vectors are said to have a *positive* orientation, if the triple-scalar product is negative, the vectors have a *negative* orientation.
-""", title="Orientation")
-```
+!!! note "Orientation"
+	The triple-scalar product, $\vec{u}\cdot(\vec{v}\times\vec{w})$, gives the volume of the parallelepiped up to sign. If the sign of this is positive, the ``3`` vectors are said to have a *positive* orientation, if the triple-scalar product is negative, the vectors have a *negative* orientation.
 
 
 #### Algebraic properties
@@ -1046,8 +1043,8 @@ Are `v` and `w` orthogonal?
 
 ```julia; hold=true; echo=false
 u,v,w = [1,2,3], [4,3,2], [5,2,1]
-ans = dot(v,w) == 0
-yesnoq(ans)
+answ = dot(v,w) == 0
+yesnoq(answ)
 ```
 
 Find the angle between `u` and `w`:
@@ -1068,8 +1065,8 @@ choices = [
 "`[-1, 6, -7]`",
 "`[-4, 14, -8]`"
 ]
-ans = 1
-radioq(choices, ans)
+answ = 1
+radioq(choices, answ)
 ```
 
 Find the area of the parallelogram formed by `v` and `w`
@@ -1114,8 +1111,8 @@ choices = [
 "An object of type `Base.Iterators.Zip` that is only realized when used",
 "A vector of values `[(1, 5), (2, 4), (3, 2)]`"
 ]
-ans = 1
-radioq(choices, ans)
+answ = 1
+radioq(choices, answ)
 ```
 
 What does `prod.(zip(u,v))` return?
@@ -1125,8 +1122,8 @@ choices = [
 "A vector of values `[5, 8, 6]`",
 "An object of type `Base.Iterators.Zip` that when realized will produce a vector of values"
 ]
-ans = 1
-radioq(choices, ans)
+answ = 1
+radioq(choices, answ)
 ```
 
 ###### Question
@@ -1143,8 +1140,8 @@ choices = [
 "``0``",
 "Can't say in general"
 ]
-ans = 1
-radioq(choices, ans)
+answ = 1
+radioq(choices, answ)
 ```
 
 ###### Question
@@ -1167,8 +1164,8 @@ choices = [
 "``x + 2y + z = 0``",
 "``x + 2y + 3z = 6``"
 ]
-ans = 1
-radioq(choices, ans)
+answ = 1
+radioq(choices, answ)
 ```
 
 
@@ -1181,8 +1178,8 @@ choices = [
 " ``\\vec{v}`` is in plane ``P_1``, as it is orthogonal to ``\\vec{n}_1`` and ``P_2`` as it is orthogonal to ``\\vec{n}_2``, hence it is parallel to both planes.",
 " ``\\vec{n}_1`` and ``\\vec{n_2}`` are unit vectors, so the cross product gives the projection, which must be orthogonal to each vector, hence in the intersection"
 ]
-ans = 1
-radioq(choices, ans)
+answ = 1
+radioq(choices, answ)
 ```
 
 ###### Question
@@ -1195,8 +1192,8 @@ choices = [
 "``\\langle 12, 12 \\rangle``",
 "``12 \\langle 1, 0 \\rangle``"
 ]
-ans = 1
-radioq(choices, ans)
+answ = 1
+radioq(choices, answ)
 ```
 
 If the vector to 3 o'clock is removed, (call this $\langle 1, 0 \rangle$) what expresses the sum of *all* the remaining vectors?
@@ -1207,8 +1204,8 @@ choices = [
 "``\\langle 1, 0 \\rangle``",
 "``\\langle 11, 11 \\rangle``"
 ]
-ans = 1
-radioq(choices, ans)
+answ = 1
+radioq(choices, answ)
 ```
 
 ###### Question
@@ -1221,8 +1218,8 @@ choices = [
 "``\\vec{u} + \\vec{v}``",
 "``\\vec{u}\\cdot\\vec{v} + \\vec{v}\\cdot \\vec{v}``"
 ]
-ans = 1
-radioq(choices, ans)
+answ = 1
+radioq(choices, answ)
 ```
 
 As the two are equal, which interpretation is true?
@@ -1233,8 +1230,8 @@ choices = [
 "The vector ``\\vec{w}`` must also be a unit vector",
 "the two are orthogonal"
 ]
-ans=1
-radioq(choices, ans)
+answ=1
+radioq(choices, answ)
 ```
 
 
@@ -1251,8 +1248,8 @@ choices = [
 "``\\vec{u}\\cdot\\vec{v} = 2``",
 "``\\vec{u}\\cdot\\vec{v} = -(\\vec{u}\\cdot\\vec{u} \\vec{v}\\cdot\\vec{v})``"
 ]
-ans = 1
-radioq(choices, ans)
+answ = 1
+radioq(choices, answ)
 ```
 
 
@@ -1274,8 +1271,8 @@ choices = [
 "The vectors are *orthogonal*, so these are all zero",
 "The vectors are all unit lengths, so these are all 1"
 ]
-ans = 1
-radioq(choices, ans)
+answ = 1
+radioq(choices, answ)
 ```
 
 ###### Question
@@ -1311,8 +1308,8 @@ choices = [
 "``n_1 (\\hat{v_1}\\times\\hat{N}) = n_2  (\\hat{v_2}\\times\\hat{N})``",
 "``n_1 (\\hat{v_1}\\times\\hat{N}) = -n_2  (\\hat{v_2}\\times\\hat{N})``"
 ]
-ans = 1
-radioq(choices, ans)
+answ = 1
+radioq(choices, answ)
 ```
 
 ###### Question
@@ -1334,6 +1331,6 @@ choices = [
 "``\\vec{a}``",
 "``\\vec{a} + \\vec{b} + \\vec{c}``"
 ]
-ans = 1
-radioq(choices, ans)
+answ = 1
+radioq(choices, answ)
 ```