# If veca = << 2,4,-1 >>, vecb = << 6,1,2 >> and vecc = veca - vecb, what is the angle between veca and vecc?

Jul 11, 2016

$1.306 \approx {74.83}^{\circ}$

#### Explanation:

$\vec{c} = \vec{a} - \vec{b}$

$= < 2 , 4 , - 1 > - < 6 , 1 , 2 >$

$= < - 4 , 3 , - 3 >$

Using that for two vectors $\vec{v}$ and $\vec{w}$ forming an angle $\theta$, we have $\cos \left(\theta\right) = \frac{\vec{v} \cdot \vec{w}}{| | \vec{v} | | \cdot | | \vec{w} | |}$, we can find the angle $\gamma$ between $\vec{a}$ and $\vec{c}$ by

$\cos \left(\gamma\right) = \frac{\vec{a} \cdot \vec{c}}{| | \vec{a} | | \cdot | | \vec{c} | |}$

=(<2,4,-1> * <-4,3,-3>)/(sqrt(2^2+4^2+(-1)^2)*sqrt((-4)^2+3^2+(-3)^2)

=(2(-4)+4(3)+(-1)(-3))/(sqrt(21)*sqrt(34)

$= \frac{7}{\sqrt{714}}$

$\implies \gamma = \arccos \left(\frac{7}{\sqrt{714}}\right) \approx 1.306$