# If A = <5 ,4 ,3 >, B = <2 ,-7 ,6 > and C=A-B, what is the angle between A and C?

Feb 19, 2016

0.929 radians ≈ 53.2^@

#### Explanation:

To calculate the angle between 2 vectors $\underline{a} \text{ and} \underline{c}$
Use the following formula.

$\cos \theta = \frac{\underline{a} . \underline{c}}{| \underline{a} | | \underline{c} |}$

here C = A - B = (5,4,3) - (2,-7,6) = (3,11,-3)

now $\underline{a} . \underline{c} = \left(5 , 4 , 3\right) . \left(3 , 11 , - 3\right)$

$= \left(5 \times 3\right) + \left(4 \times 11\right) + \left(3 \times - 3\right) = 15 + 44 - 9 = 50$

$| \underline{a} | = \sqrt{{5}^{2} + {4}^{2} + {3}^{2}} = \sqrt{25 + 16 + 9} = \sqrt{50}$

and $| \underline{c} | = \sqrt{{3}^{2} + {11}^{2} + {\left(- 3\right)}^{2}} = \sqrt{9 + 121 + 9} = \sqrt{139}$

rArr theta = cos^-1(50/(sqrt50xxsqrt139)) ≈ 0.929 " radians"