Are the vectors u=<1, -2> and v=<4, 8> parallel, orthogonal, or neither?

May 2, 2017

Answer:

Neither parallel nor orthogonal. The angle between the two vectors is:

$\theta \approx {127}^{\circ}$ or $2.21$radians

Explanation:

Given: $\vec{u} = < 1 , - 2 >$ and $\vec{v} = < 4 , 8 >$

Compute the dot-product:

$\vec{u} \cdot \vec{v} = \left(1\right) \left(4\right) + \left(- 2\right) \left(8\right) = - 12$

We computed the dot-product by multiplying the respective components of the vectors. Another definition of the dot-product is the magnitudes of the vectors multiplied by the angle between them:

$\vec{u} \cdot \vec{v} = | \vec{u} | | \vec{v} | \cos \left(\theta\right) \text{ [1]}$

Compute the magnitudes of the two vectors:

$| \vec{u} | = \sqrt{{1}^{2} + {\left(- 2\right)}^{2}} = \sqrt{5}$
$| \vec{v} | = \sqrt{{4}^{2} + {8}^{2}} = \sqrt{80}$

Substitute the values for the dot-product and the magnitudes into equation [1]:

$- 12 = \sqrt{5} \sqrt{80} \cos \left(\theta\right)$

$- 12 = \sqrt{400} \cos \left(\theta\right)$

$- 12 = 20 \cos \left(\theta\right)$

$\cos \left(\theta\right) = - \frac{12}{20}$

$\theta = {\cos}^{-} 1 \left(- \frac{3}{5}\right)$

$\theta \approx {127}^{\circ}$ or $2.21$radians