Question

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

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

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