Question

Determine whether the given vectors are orthogonal, parallel, or neither. u = ⟨-3, 9, 6⟩, v = ⟨4, -12, -8⟩

Vectors
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asked 2021-06-04
Determine whether the given vectors are orthogonal, parallel, or neither. u = ⟨-3, 9, 6⟩, v = ⟨4, -12, -8⟩

Expert Answers (1)

2021-06-05
First, we find a dot product of the vectors \(\vec{u}\) and \(\vec{v}\)
\(\vec{u}\cdot\vec{v}=-3\cdot4+9(-12)+6(-8)\)
\(=-12-108-48\)
\(=-168\ne0\)
Second, we find the agle between the given vectors, for which we have to find length of vectors \(\vec{u}\) and \(\vec{v}\)
\(|\vec{a}|=\sqrt{(-3)^2+9^2+6^2}\)
\(=\sqrt{9+81+36}\)
\(=\sqrt{126}\)
\(|\vec{b}|=\sqrt{4^2+(-12)^2+(-8)^2}\)
\(=\sqrt{16+144+64}\)
\(=\sqrt{224}\)
So,
\(\theta=\arccos(-\frac{168}{\sqrt{126}\sqrt{224}})\)
\(=\arccos(-1)\)
\(=180^\circ\)
\(=\pi\)
Therefore, the given vectors are parallel.
Result:
The given vectors are parallel
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