Without finding the angle between two vectors, we can determine if they are parallel if they are scalar multiples of each other. If \(\displaystyle{u}={k}{v}{\quad\text{or}\quad}{v}={k}{u}\) for non-zero value of k since we are simply scaling. The two vectors are orthogonal if their dot product is 0: \(\displaystyle{u}\times{v}={0}\)

Check if they are parallel:

Since \(\displaystyle{v}\ne{5}{u},⟨{20},{15}⟩\ne{5}⟨-{3}.{4}⟩\)

Then u and v are not parallel.

Check if they are orthogonal:

Find the dot product:

\(\displaystyle{u}\times{v}={\left(-{3}\right)}{\left({20}\right)}+{\left({4}\right)}{\left({15}\right)}=-{60}+{60}={0}\)

Since \(\displaystyle{u}\times{v}={0}\), then u and v are orthogonal.

Check if they are parallel:

Since \(\displaystyle{v}\ne{5}{u},⟨{20},{15}⟩\ne{5}⟨-{3}.{4}⟩\)

Then u and v are not parallel.

Check if they are orthogonal:

Find the dot product:

\(\displaystyle{u}\times{v}={\left(-{3}\right)}{\left({20}\right)}+{\left({4}\right)}{\left({15}\right)}=-{60}+{60}={0}\)

Since \(\displaystyle{u}\times{v}={0}\), then u and v are orthogonal.