# How do you use the definition of the scalar product, find the angles between the following pairs of vectors: A = - 2j -5k and B = - 4i + j - 4k?

Jun 3, 2017

The angle is =54.4º

#### Explanation:

The dot product or scalar product of 2 vectors, $\vec{A}$ and $\vec{B}$ is

$\vec{A} . \vec{B} = | | \vec{A} | | \cdot | | \vec{B} | | \cdot \cos \theta$

Where $\theta$ is the angle between the 2 vectors

$\cos \theta = \frac{\vec{A} . \vec{B}}{| | \vec{A} | | \cdot | | \vec{B} | |}$

The dot product is $= \vec{A} . \vec{B} = < 0 , - 2 , - 5 > . < - 4 , 1 , - 4 >$

$= \left(0 \cdot - 4\right) + \left(- 2 \cdot 1\right) + \left(- 5 \cdot - 4\right)$

$= 0 - 2 + 20 = 18$

The magnitude of $\vec{A}$ is $= | | < 0 , - 2 , - 5 > | | = \sqrt{0 + 4 + 25} = \sqrt{29}$

The magnitude of $\vec{B}$ is $= | | < - 4 , 1 , - 4 > | | = \sqrt{16 + 1 + 16} = \sqrt{33}$

$\cos \theta = \frac{18}{\sqrt{29} \cdot \sqrt{33}} = 0.58$

theta=54.4º