# What is the angle between <7 , -2 , -4 >  and  < 5 , -2 , 1 > ?

Nov 12, 2016

$\theta \approx 0.69 \text{ radians}$

#### Explanation:

Compute the dot-product:

$< 7 , - 2 , - 4 > \cdot < 5 , - 2 , 1 > = \left(7\right) \left(5\right) + \left(- 2\right) \left(- 2\right) + \left(- 4\right) \left(1\right)$

$< 7 , - 2 , - 4 > \cdot < 5 , - 2 , 1 > = 35$

Compute the magnitude of both vectors:

$| < 7 , - 2 , - 4 > | = \sqrt{{7}^{2} + {\left(- 2\right)}^{2} + {\left(- 4\right)}^{2}}$

$| < 7 , - 2 , - 4 > | = \sqrt{69}$

$| < 5 , - 2 , 1 > | = \sqrt{{5}^{2} + {\left(- 2\right)}^{2} + {1}^{2}}$

$| < 5 , - 2 , 1 > | = \sqrt{30}$

The other way to compute the dot-product is:

$< 7 , - 2 , - 4 > \cdot < 5 , - 2 , 1 > = | < 7 , - 2 , - 4 > | | < 5 , - 2 , 1 > | \cos \left(\theta\right)$

where $\theta$ is the angle between the two vectors.

Substitute the values that we have computed:

$35 = \sqrt{69} \sqrt{30} \cos \left(\theta\right)$

Solve for $\theta$:

$\theta = {\cos}^{-} 1 \left(\frac{35}{\sqrt{\left\{69\right\} \left\{30\right\}}}\right)$

$\theta \approx 0.69 \text{ radians}$