Update appendix-e.md

Author: pavly-gerges

calculus/appendix-e.md

@@ -5,8 +5,8 @@
 - Recall, 2 intersecting vectors $u$ and $v$ together with the resultant vetcor $w$; such that, $$w = u - v$$
 
 - From the "Law of Cosines":
-$$Since, ||w||^2 = ||u||^2 + ||v||^2 - 2 * ||u|| * ||v|| * cos(\theta)$$
-$$Then, cos(\theta) = (||w||^2 - ||u||^2 - ||v||^2) / -2 * ||u|| * ||v||\ Lemma\.01$$ 
+$$Since, ||w||^2 = ||u||^2 + ||v||^2 - 2 * ||u||^2 * ||v||^2 * cos(\theta)$$
+$$Then, cos(\theta) = (||w||^2 - ||u||^2 - ||v||^2) / -2 * ||u||^2 * ||v||^2\ Lemma\.01$$ 
 
 $$ ------- $$
 
@@ -18,14 +18,23 @@ $$||u|| = \sqrt{{u_x}^2 + {u_y}^2 + {u_z}^2}$$
 
 $$||w|| = ||u-v|| = \sqrt{{(u_x - v_x)}^2 + {(u_y - v_y)}^2 + {(u_z - v_z)}^2}$$
 
-$$Then, 1)\ ||v||^2 = {v_x}^2 + {v_y}^2 + {v_z}^2$$
+$$Then,\ Lemma.02:$$
 
-$$ 2)\ ||u||^2 = {u_x}^2 + {u_y}^2 + {u_z}^2$$
+$$1)\ ||v||^2 = {v_x}^2 + {v_y}^2 + {v_z}^2$$
 
-$$ 3)\ ||w||^2 = {u_x - v_x}^2 + {u_y - v_y}^2 + {u_z - v_z}^2$$
+$$2)\ ||u||^2 = {u_x}^2 + {u_y}^2 + {u_z}^2$$
+
+$$3)\ ||w||^2 = {u_x - v_x}^2 + {u_y - v_y}^2 + {u_z - v_z}^2$$
 
 $$ ------- $$
 
+- By back-substitution in $Lemma.01$:
+
+$$cos(\theta) = (||w||^2 - ||u||^2 - ||v||^2) / -2 * ||u||^2 * ||v||^2 = {(R.H.S)}_1 / {(R.H.S)}_2
+
+$${(R.H.S)}_1 = (||w||^2 - ||u||^2 - ||v||^2)$$
+$${(R.H.S)}_2 = -2 * ||u||^2 * ||v||^2 = -2 * ({u_x}^2 + {u_y}^2 + {u_z}^2) * ({v_x}^2 + {v_y}^2 + {v_z}^2)
+
 $$Hence, cos(\theta) = u.v / ||u|| * ||v||$$
 $$Thence, u.v = ||u|| * ||v|| * cos(\theta)$$