# How do you determine whether u and v are orthogonal, parallel or neither given u=<3, 15> and v=<-1, 5>?

Dec 13, 2016

#### Explanation:

Compute the dot-product:

$\overline{u} \cdot \overline{v} = 3 \left(- 1\right) + 15 \left(5\right) = 72$

The two vectors are not orthogonal; we know this, because orthogonal vectors have a dot-product that is equal to zero.

Determine whether the two vectors are parallel by finding the angle between them.

Compute the magnitude of both vectors:

$| | \overline{u} | | = \sqrt{{3}^{2} + {15}^{2}} = \sqrt{234}$

$| | \overline{v} | | = \sqrt{{\left(- 1\right)}^{2} + {5}^{2}} = \sqrt{26}$

The angle between them is:

$\theta = {\cos}^{-} 1 \left(\frac{72}{\sqrt{234} \sqrt{26}}\right)$

$\theta \approx {22.6}^{\circ}$

If they were parallel the angle would be ${0}^{\circ} \mathmr{and} {180}^{\circ}$, therefore, the two vectors are not parallel.

Dec 13, 2016

The vectors are not parallel and not orthogonal.

#### Explanation:

To see if 2 vectors, we do a dot product

vecu.vecv=〈3,15〉.〈-1,5〉=3*-1+15*5=-3+75=72

As the dot product is $\ne 0$, the vectors are not perpendicular

If 2 vectors are parallel,

then, $\vec{u} = k \vec{v}$

〈3,15〉=k〈-1,5〉

$3 = - k$ and $15 = 5 k$

Therefore, $k = - 3$ and $k = 3$

This is not possible, so the vectors are not parallel