# If A = <4 ,-5 ,1 >, B = <5 ,1 ,-3 > and C=A-B, what is the angle between A and C?

Jun 20, 2016

Reqd. angle $= \arccos \left(\frac{30}{\sqrt{2226}}\right) .$

#### Explanation:

$A = < 4 , - 5 , 1 > , B = < 5 , 1 , - 3 >$
$\therefore C = A - B = < 4 - 5 , - 5 - 1 , 1 - \left(- 3\right) >$ = $< - 1 , - 6 , 4 > .$

Let the reqd. angle be $\theta$. Then, using dot product of $A$ & $C$, we have,
$A . C = | | A | | \cdot | | C | \setminus \cos \theta \Rightarrow < 4 , - 5 , 1 > . < - 1 , - 6 , 4 >$=$\sqrt{{4}^{2} + {\left(- 5\right)}^{2} + {1}^{2}} \cdot \sqrt{{\left(- 1\right)}^{2} + {\left(- 6\right)}^{2} + {4}^{2}} \cdot \cos \theta$

$\therefore - 4 + 30 + 4 = \sqrt{42} \cdot \sqrt{53} \cdot \cos \theta$
$\therefore 30 = \sqrt{2226} \cdot \cos \theta$
$\therefore \cos \theta = \frac{30}{\sqrt{2226}}$
$\therefore \theta = \arccos \left(\frac{30}{\sqrt{2226}}\right) .$