Update appendix-e.md

Author: pavly-gerges

calculus/appendix-e.md

@@ -2,7 +2,14 @@
 
 ## 1) The dot product: 
 
-$$u.v = ||u|| * ||v|| * cos(\theta)$$
+- Recall, 2 intersecting vectors $u$ and $v$ together with the resultant vetcor $w$; such that, $$w = u + v$$
+
+- From the "Law of Cosines":
+$$Since, ||w||^2 = ||u||^2 + ||v||^2 - 2 * ||u|| * ||v|| * cos(\theta)$$
+$$cos(\theta) = (||w||^2 - ||u||^2 - ||v||^2) / -2 * ||u|| * ||v||$$
+
+$$Hence, cos(\theta) = u.v / ||u|| * ||v||$$
+$$Thence, u.v = ||u|| * ||v|| * cos(\theta)$$
 
 ## 2) Projection vectors: