Angle Between Two Lines (Direction Ratios)

Angle between two lines with direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂). Derived by normalizing DRs to DCs first.

Given DRs $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$, the DCs are:

$$\hat{u}_1 = \frac{(a_1,b_1,c_1)}{\sqrt{a_1^2+b_1^2+c_1^2}}, \quad \hat{u}_2 = \frac{(a_2,b_2,c_2)}{\sqrt{a_2^2+b_2^2+c_2^2}}$$ $$\cos\theta = \hat{u}_1\cdot\hat{u}_2 = \frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\cdot\sqrt{a_2^2+b_2^2+c_2^2}}$$

Taking the acute angle, use the absolute value of the numerator.

Perpendicularity condition: $a_1a_2+b_1b_2+c_1c_2 = 0$.

Parallelism condition: $a_1/a_2 = b_1/b_2 = c_1/c_2$.

Example: Angle between lines with DRs $(1, 2, 2)$ and $(3, 4, 0)$:

$$\cos\theta = \frac{|3+8+0|}{\sqrt{9}\cdot\sqrt{25}} = \frac{11}{15}$$