# If A = <1 ,6 ,9 >, B = <-9 ,-6 ,-8 > and C=A-B, what is the angle between A and C?

Mar 11, 2016

${20.438}^{\circ}$

#### Explanation:

$C = A - B = \left(1 , 6 , 9\right) - \left(- 9 , - 6 , - 8\right)$

$= \left(10 , 12 , 17\right)$.

Hence the angle $\theta$ between $A \mathmr{and} C$ may be given in terms of the Euclidean inner product as follows

$\theta = {\cos}^{- 1} \left(\frac{A \cdot C}{| | A | | | | C | |}\right)$

$= {\cos}^{- 1} \left(\frac{1 \times 10 + 6 \times 12 + 9 \times 17}{\sqrt{{1}^{2} + {6}^{2} + {9}^{2}} \left(\sqrt{{10}^{2} + {12}^{2} + {17}^{2}}\right)}\right)$

$= {\cos}^{- 1} \left(\frac{235}{\sqrt{118} \sqrt{533}}\right)$

$= {20.438}^{\circ}$