# If A = <2 ,4 ,-9 >, B = <3 ,1 ,-5 > and C=A-B, what is the angle between A and C?

Nov 4, 2016

The angle is 26.1º

#### Explanation:

To determine the angle $\theta$ two vectors, we use the dot product definition:
vecA.vecC=∥vecA∥*∥vecC∥costheta
Where ∥vecA∥ is the modulus of vector $\vec{A}$

So, costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)
vecC=vecA-vecB=〈-1,3,-4〉
The dot product =〈2,4,-9〉.〈-1,3,-4〉=-2+12+36=46

Modulus of $\vec{A} = \sqrt{4 + 16 + 81} = \sqrt{101}$
Modulus of $\vec{C} = \sqrt{1 + 9 + 16} = \sqrt{26}$
$\cos \theta = \frac{46}{\sqrt{101} \cdot \sqrt{26}} = 0.898$
theta=26.1º