Update appendix-e.md

Author: pavly-gerges

calculus/appendix-e.md

@@ -9,8 +9,19 @@ $$Since, ||w||^2 = ||u||^2 + ||v||^2 - 2 * ||u|| * ||v|| * cos(\theta)$$
 $$Then, cos(\theta) = (||w||^2 - ||u||^2 - ||v||^2) / -2 * ||u|| * ||v||\ Lemma\.01$$ 
 
 $$Since, ||w|| = \sqrt{(u_x - v_x)^2 + (u_y - v_y)^2 + (u_z - v_z)^2}$$
+
 $$||v|| = \sqrt{{v_x}^2 + {v_y}^2 + {v_z}^2}$$
-$$||u|| = \sqrt{$$
+
+$$||u|| = \sqrt{{u_x}^2 + {u_y}^2 + {u_z}^2}$$
+
+$$||w|| = ||u-v|| = \sqrt{{u_x - v_x}^2 + {u_y - v_y}^2 + {u_z - v_z}^2}$$
+
+$$Then, 1)\ ||v||^2 = {v_x}^2 + {v_y}^2 + {v_z}^2$$
+
+$$ 2)\ ||u||^2 = {u_x}^2 + {u_y}^2 + {u_z}^2$$
+
+$$ 3)\ ||w||^2 = {u_x - v_x}^2 + {u_y - v_y}^2 + {u_z - v_z}^2$$
+
 
 $$Hence, cos(\theta) = u.v / ||u|| * ||v||$$
 $$Thence, u.v = ||u|| * ||v|| * cos(\theta)$$