Condition for Parallel Lines

Two lines are parallel iff their direction ratios are proportional. The angle between them is 0°.

Two lines are parallel iff $\theta = 0°$, i.e. $\cos\theta = 1$:

$$\frac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\cdot\sqrt{a_2^2+b_2^2+c_2^2}} = 1$$

By the Cauchy-Schwarz equality condition, this holds iff $(a_1,b_1,c_1) = k(a_2,b_2,c_2)$ for some scalar $k$:

$$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$

In vector form: $\mathbf{b_1} = k\mathbf{b_2}$ (direction vectors are parallel).

Parallel vs coincident: Two lines can be parallel (distinct, never meeting) or coincident (the same line). Both have proportional DRs. To distinguish: check if a point of one line lies on the other.

Parallel and coplanar: Two parallel lines always lie in a common plane (they determine a unique plane). Two skew lines cannot be parallel.