Question

Determine whether the vectors u and v are parallel, orthogonal, or neither.

Vectors
ANSWERED
asked 2021-08-18
Determine whether the vectors u and v are parallel, orthogonal, or neither.
\(\displaystyle{u}=⟨−{3},{4}⟩,{v}−⟨\frac{{20}}{{15}}⟩\)
\(\displaystyle{u}=⟨−{3},{4}⟩,{v}−⟨\frac{{20}}{{15}}⟩\)

Expert Answers (1)

2021-08-19
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.
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