Question

State if the two vectors are parallel, orthogonal, or neither.

u=18i+8j

v=9i+4j

Answer 1

Answer:

Parallel, since $\vec u = 2\cdot \vec v$.

Step-by-step explanation:

The relation between both vectors is determined by the use of the dot product, whose expression is:

$\cos \theta = \frac{\vec u \bullet \vec v}{\|\vec u\| \|\vec v\|}$

Where:

$\cos \theta = 1$ if vectors are parallel to each other and $\cos \theta = 0$ if vectors are orthogonal. Then, norms and dot product are calculated hereafter:

$\|\vec u\| = \sqrt{18^{2}+8^{2}}$

$\|\vec u\| \approx 19.698$

$\|\vec v \| = \sqrt{9^{2}+4^{2}}$

$\|\vec v\| \approx 9.849$

$\vec u \bullet \vec v = (18)\cdot (9) + (8)\cdot (4)$

$\vec u \bullet \vec v = 194$

$\cos \theta = \frac{194}{(19.698)\cdot (9.849)}$

$\cos \theta = 1$

The two vectors are parallel to each other, which is also supported by the fact that one vector is multiply of the other one. That is,

$18i + 8j = 2\cdot (9i + 4j)$

$\vec u = 2\cdot \vec v$