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

\(\vec{u}\cdot\vec{v}=-3\cdot4+9(-12)+6(-8)\)

\(=-12-108-48\)

\(=-168\ne0\)

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