From 08bf69e67cd1aee49c9f62147713dd781d217dbf Mon Sep 17 00:00:00 2001
From: mlamich1 <139090918+mlamich1@users.noreply.github.com>
Date: Fri, 26 Jul 2024 18:06:50 +0000
Subject: [PATCH 1/2] Added_Angle_between_Two_Vectors_Did
---
linear-algebra/source/future-ON/01.ptx | 30 +++++++++++++++++++++++---
1 file changed, 27 insertions(+), 3 deletions(-)
diff --git a/linear-algebra/source/future-ON/01.ptx b/linear-algebra/source/future-ON/01.ptx
index 7fcdf3835..250844bcf 100644
--- a/linear-algebra/source/future-ON/01.ptx
+++ b/linear-algebra/source/future-ON/01.ptx
@@ -148,8 +148,32 @@ The dot product combines two vectors and creates a scalar that gives us geometri
Need to create a segue to the fact that you can find the angle between vectors using the Law of Cosines as a starting place.
-
-
+
+
+ Given two vectors \vec u and \vec v in \IR^n, such that \vec u and \vec v are not parallel, let \theta be the angle between the two vectors,
+ then
+
+ \cos \theta = \frac{\vec u \cdot \vec v }{|\vec u||\vec v|}.
+
+ By Law of Cosine:
+
+ |\vec u- \vec v|^2 = |\vec u|^2+|\vec v|^2- 2 |\vec u||\vec v| \cos \theta.
+
+ Using dot product of vector \vec u-\vec v with itself:
+
+
+ |\vec u- \vec v|^2 \amp = (\vec u- \vec v) \cdot (\vec u- \vec v) \amp
+ \amp = \vec u \cdot \vec u - 2 ( \vec u \cdot \vec v) + \vec v \cdot \vec v \amp
+ \amp = |\vec u|^2 - 2 ( \vec u.\vec v) + |\vec v|^2 \amp
+
+Hence from above, we have:
+
+ \vec u.\vec v = |\vec u||\vec v| \cos \theta .
+
+
+
+
+ One of these activity should be deleted from here.
@@ -165,7 +189,7 @@ The dot product combines two vectors and creates a scalar that gives us geometri
Consider two vectors \vec u =\left[\begin{array}{c} 1 \\ 3 \\ 4 \\ -4 \end{array}\right] and \vec v = \left[\begin{array}{c} -1 \\ -3 \\ -4 \\ 4 \end{array}\right].
- Use dot product to determine ||\vec u|| and ||\vec v||.
+ Use dot product to determine |\vec u| and |\vec v|.
Using dot product, find the distance between \vec u and \vec v .
Find the angle between \vec u and \vec v.
From 447e3f7c27e3827abad8c11ac9083bf1f0a4d699 Mon Sep 17 00:00:00 2001
From: mlamich1 <139090918+mlamich1@users.noreply.github.com>
Date: Fri, 26 Jul 2024 19:28:05 +0000
Subject: [PATCH 2/2] DotProduct_afterchange
---
linear-algebra/source/future-ON/01.ptx | 4 ++--
1 file changed, 2 insertions(+), 2 deletions(-)
diff --git a/linear-algebra/source/future-ON/01.ptx b/linear-algebra/source/future-ON/01.ptx
index 250844bcf..471381289 100644
--- a/linear-algebra/source/future-ON/01.ptx
+++ b/linear-algebra/source/future-ON/01.ptx
@@ -155,7 +155,7 @@ The dot product combines two vectors and creates a scalar that gives us geometri
\cos \theta = \frac{\vec u \cdot \vec v }{|\vec u||\vec v|}.
- By Law of Cosine:
+ By the Law of Cosine:
|\vec u- \vec v|^2 = |\vec u|^2+|\vec v|^2- 2 |\vec u||\vec v| \cos \theta.
@@ -168,7 +168,7 @@ The dot product combines two vectors and creates a scalar that gives us geometri
Hence from above, we have:
- \vec u.\vec v = |\vec u||\vec v| \cos \theta .
+ \vec u \cdot \vec v = |\vec u||\vec v| \cos \theta .