commit 2b610dd03c0ac12b0dc25e269cad656d4c85101e
parent 6af8713e6ac446cb48b48263603019513efda807
Author: Andrew <andrewlaack1@gmail.com>
Date: Mon, 8 Jul 2024 19:44:10 -0500
Completed notes
Diffstat:
12 files changed, 112 insertions(+), 3 deletions(-)
diff --git a/AngleBetweenVectors.md b/AngleBetweenVectors.md
@@ -13,5 +13,6 @@ Khan
2. Take dot product divided by lengths of vectors to find cosine of the angle (solve)
- cos(theta) = (u dot v)/(||u||||v||)
+## Intuition
+
-## Usage
diff --git a/Arccos.md b/Arccos.md
@@ -0,0 +1,8 @@
+:trig:
+# Arccos
+
+SS
+
+## Notes
+
+**Definition:** Arccos is the inverse of cosine.
diff --git a/Arcsin.md b/Arcsin.md
@@ -0,0 +1,8 @@
+:trig:
+# Arcsin
+
+SS
+
+## Notes
+
+**Definition:** Arcsin is the inverse of sine.
diff --git a/CrossProduct.md b/CrossProduct.md
@@ -19,3 +19,7 @@ b = [b_1, b_2, b_3]
o = [a_2b_3 - a_3b_2 , a_3b_1 - a_1b_3 , a_1b_2 - a_2b_1]
Conceptually, we take the determinant of the bottom two rows then reverse determinent of top and bottom rows then deteminent of top two rows.
+
+The length of the cross product is the distance the second vector is projected onto the first vector multiplied by the length of the first vector. This is pretty much the inversion of the dot product's value where we find the projected distance of the second multiplied by the distance of the first.
+
+The cross product's length is 0 if both vectors are colinear. Conversely, it is maximized when the vectors are orthogonal.
diff --git a/DistanceToPlane.md b/DistanceToPlane.md
@@ -0,0 +1,16 @@
+:lin-alg:
+# Distance to Plane
+
+Distance from arbitrary point to plane
+
+## Notes
+
+If we take any point on the plane and then find the length of the opposite side of the new right triangle we then have the distance from the plane to the point.
+
+As such this is as simple as taking the dot product of the normal vector and the vector that connects the representative point and the other point. We then divide this by the lenght of the normal vector and that is the lenght of the opposite side which is also the distance between the plane and the point.
+
+To find distance to plane from point do the following:
+
+1. Find the vector from a representative point to the point
+2. Calculate the dot product between this vector connecting these two points and the normal vector
+3. Divide this value by the length of the normal vector
diff --git a/DotProduct.md b/DotProduct.md
@@ -15,4 +15,17 @@ This value is zero if the vectors are orthogonal.
The dotproduct in a geometric sense is u dot v = ||u|| ||v|| cos(theta) where theta is the angle between the two vectors. As such, when the angle between them is greater than 90 and less than 270 we find the dot product is negative.
-This intuition leads to how we can find the angle between two vectors namely theta above.
+
+### Intuition Of DP
+
+By definition, the dot product can be stated as follows where || defines lenghts of vectors and theta is the angle between the two vectors:
+
+a dot b = ||a|| ||b|| cos(theta)
+
+The dot product is the projection of the second vector onto the first vector multiplied by the length of the first vector where the first vector is the one that has a lesser angle.
+
+Based on this information, it is intuitive that when two vectors are almost orthogonal the dot product is very small. Conversely, when two vectors are almost dependent, the dot product will then be large.
+
+This explains why orthogonal vectors have a dot product of zero.
+
+**Product of the lengths of the vectors that are moving in the same direction.**
diff --git a/LinearAlgebra.md b/LinearAlgebra.md
@@ -26,4 +26,10 @@ The basis of linear algebra is solving systems of equations.
[[LawOfCosines.md]]
[[EquationOfAPlane.md]]
[[CrossProduct.md]]
-
+[[Arcsin.md]]
+[[Arccos.md]]
+[[TripleProductExpansion.md]]
+[[NormalVector.md]]
+[[DistanceToPlane.md]]
+[[PlaneToPlaneDistance.md]]
+[[ReducedRowEchelonForm.md]]
diff --git a/Matrix.md b/Matrix.md
@@ -16,3 +16,12 @@ Matricies can be used to describe systems of equations as follows:
[-2 1] [y] = [3]
This is the form because we distribute x on the first column and y on the second column.
+
+
+## Matrix Vector Product
+
+The product of a matrix and a vector is another vector of the same size as the original vector. This assumes the number of components in the vector is the same as the number of columns in the matrix.
+
+[a b c] [j] [aj bk cl]
+[d e f] @ [k] = [dj ek fl] (note the result is a vector not a matrix)
+[g h i] [l] [gj hk il]
diff --git a/NormalVector.md b/NormalVector.md
@@ -0,0 +1,8 @@
+:lin-alg:
+# Normal Vector
+
+Khan
+
+## Notes
+
+**Definition:** The normal vector of a hyperplane is a vector that is orthogonal to the hyperplane (there are infinitely many as this is simply a direction and the magnitude does not matter unless specifying unit normal vector).
diff --git a/PlaneToPlaneDistance.md b/PlaneToPlaneDistance.md
@@ -0,0 +1,18 @@
+:lin-alg:
+# Plane to Plane Distance
+
+Khan
+
+## Notes
+
+See [[DistanceToPlane.md]] for distance from plane to point.
+
+This only is useful for planes that are paralell otherwise they will intersect.
+
+Steps:
+
+1. Find equation of both planes
+2. Find representative point
+3. Take [[DistanceToPlane.md]] from the rep point to the other plane.
+
+This is true because all points will be the same distance from the other plane given that they are paralell. Otherwise, it would be imperative that they intercect and thus have a min distance of 0.
diff --git a/ReducedRowEchelonForm.md b/ReducedRowEchelonForm.md
@@ -0,0 +1,8 @@
+:lin-alg:
+# Reduced Row Echelon Form
+
+Khan
+
+## Notes
+
+**Definition:** Reduced row echelon form is a form of matrix where each row has a 1 after the zeoes that are all on the left side of the one. Additionally, each row above another row must have its 1 further to the left than the prior one, and all values to the right of the one should be zeroes if possible.
diff --git a/TripleProductExpansion.md b/TripleProductExpansion.md
@@ -0,0 +1,10 @@
+:lin-alg:
+# Triple Product Expansion
+
+Khan
+
+## Notes
+
+**Definition:** The triple product expansion states the combined cross product of three vectors a x (b x c) = b(a dot c) - c(a dot b)
+
+This is also known as lagrange's formula.
