Update appendix-e.md

calculus/appendix-e.md

@@ -69,35 +69,35 @@ the one we would like to utilize its direction to find the projection vector.
 
 1) Finding the norm of the vector component $||\vec{u_x}||$ that is coincident to the projectile vector $\vec{v}$:
 
-$$Since, cos(\theta) = ||\vec{u_x}|| / ||\vec{u}||$$
+$$Since,cos(\theta)=||\vec{u_x}||/||\vec{u}||$$
 
-$$Then, ||\vec{u_x}|| = ||\vec{u}|| * cos(\theta)$$
+$$Then,||\vec{u_x}||=||\vec{u}||*cos(\theta)$$
 
 2) Finding the vector norm $||\vec{v}||$ of the projectile vector $\vec{v}$:
 
-$$||\vec{v}|| = \sqrt{\{v_x}^2 + {v_y}^2 + {v_z}^2\}$$
+$$||\vec{v}||=\sqrt{\{v_x}^2+{v_y}^2+{v_z}^2\}$$
 
-3) Finding the normalization ratio using the scalar division property $\vec{v_{unit}} = \vec{v} / ||\vec{v}||$ of the projectile vector (base vector).
+3) Finding the normalization ratio using the scalar division property $\vec{v_{unit}}=\vec{v}/||\vec{v}||$ of the projectile vector (base vector).
 
-4) Using the scalar multiplication property $||\vec{u_x}|| * \vec{v_{unit}}$:
+4) Using the scalar multiplication property $||\vec{u_x}||*\vec{v_{unit}}$:
 
 > $Lemma.01$
 
-$$||\vec{u_x}|| * \vec{v_{unit} = (||\vec{u}|| * cos(\theta)) * \vec{v_{unit}$$
+$$||\vec{u_x}||*\vec{v_{unit}}=(||\vec{u}||*cos(\theta))\*\vec{v_{unit}}$$
 
-$$Since, \vec{u}.\vec{v} = ||\vec{u}|| * ||\vec{v}|| * cos(\theta)$$
+$$Since,\vec{u}.\vec{v}=||\vec{u}||*||\vec{v}||\*cos(\theta)$$
 
 > $Lemma.02$
 
-$$Then, ||\vec{u}|| * cos(\theta) = (\vec{u}.\vec{v}) / ||\vec{v}||$$
+$$Then,||\vec{u}||*cos(\theta)=(\vec{u}.\vec{v})/||\vec{v}||$$
 
 5) Then, from $Lemma.01$ and $Lemma.02$, we can deduce the projection vector formula in terms of the dot product between 2 vectors as follows: 
 
-$$\vec{proj_{\vec{v}}} \vec{u} = (\vec{u}.\vec{v} * \vec{v_{unit}) / ||\vec{v}||$$
+$$\vec{proj_{\vec{v}}}\vec{u}=(\vec{u}\.\vec{v}\*\vec{v_{unit})/||\vec{v}||$$
 
 6) Another formula when $\vec{v_{unit}}$ is broken: 
 
-$$\vec{proj_{\vec{v}}} \vec{u} = (\vec{u}.\vec{v} * \vec{v}) / ||\vec{v}||^2$$
+$$\vec{proj_{\vec{v}}}\vec{u}=(\vec{u}.\vec{v}*\vec{v})/||\vec{v}||^2$$
 
 > Note:
 > * This is could be applied to other components of vector $\vec{u}$, the $\vec{u_y}$, and the $\vec{u_z}$, and any vector could be utilized as the base or the projectile vector.