diff --git a/linear-algebra/source/future-ON/01.ptx b/linear-algebra/source/future-ON/01.ptx
index 7fcdf3835..471381289 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 the 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 \cdot \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.