# Question #3aaaa

Apr 2, 2017

A. Parallel

#### Explanation:

Given: $\vec{v} = 3 \hat{i} + 2 \hat{j}$, $\vec{w} = 6 \hat{i} + 4 \hat{j}$

Compute the Dot Product :

$\vec{v} \cdot \vec{w} = 3 \left(6\right) + \left(2\right) \left(4\right)$

$\vec{v} \cdot \vec{w} = 26$

The Dot Product is not 0, therefore, the vectors are not Orthogonal

Compute the magnitudes of both vectors:

$| \vec{v} | = \sqrt{{3}^{2} + {2}^{2}}$

$| \vec{v} | = \sqrt{9 + 4}$

$| \vec{v} | = \sqrt{13}$

$| \vec{w} | = \sqrt{{6}^{2} + {4}^{2}}$

$| \vec{w} | = \sqrt{36 + 16}$

$| \vec{w} | = \sqrt{52}$

Use an alternative definition of the Dot Product:

$\vec{v} \cdot \vec{w} = | \vec{v} | | \vec{w} | \cos \left(\theta\right)$

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

Substitute in the known values:

$26 = \sqrt{13} \sqrt{52} \cos \left(\theta\right)$

$26 = \sqrt{676} \cos \left(\theta\right)$

$26 = 26 \cos \left(\theta\right)$

$\cos \left(\theta\right) = 1$

$\theta = 0$

This indicates that the two vectors are parallel.