# What is the angle between <6 , 5 , 7 >  and  < 3 , -4 , 8 > ?

##### 1 Answer
Mar 16, 2018

The angle between the vectors color(green)(<6 , 5 , 7 > and < 3 , -4 , 8 >  is given by color(blue)(56.92^@ approx.

#### Explanation:

Given:

color(red)(vec a=6i + 5j + 7k

color(red)(vec b=3i - 4j + 8k

color(green)(cos theta = (vec a*vec b)/(|a|*|b|)

rArr [(6*3)+(5)*(-4) + (7)(8)]/[sqrt(6^2+5^2+7^2)*sqrt(3^2+(-4)^2+8^2]

rArr [18-20 + 56]/[sqrt(36+25+49)*sqrt(9+16+64]

$\Rightarrow \frac{54}{\sqrt{110} \cdot \sqrt{89}}$

$\Rightarrow \frac{54}{\left(10.48809\right) \cdot \left(9.433981\right)} \approx \left(\frac{54}{98.94443}\right)$

Hence,

color(green)(cos theta ~~ (0.545760895)

To find the required angle, we must find the value of color(green)(theta.

$\theta = {\cos}^{-} 1 \left(0.545760895\right)$

Hence,

$\theta \approx 56.92332412$

The angle between the vectors color(green)(<6 , 5 , 7 > and < 3 , -4 , 8 >  is given by color(blue)(56.92^@ approx.

Hope it helps.