Condition for Perpendicular Lines

Two lines with direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂) are perpendicular iff their dot product is zero. Equivalently l₁l₂+m₁m₂+n₁n₂ = 0.

Two lines are perpendicular iff the angle between them is 90°, i.e. $\cos\theta = 0$.

From the angle formula:

$$\cos\theta = \frac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\cdot\sqrt{a_2^2+b_2^2+c_2^2}} = 0$$

The denominator is never zero (for non-degenerate lines), so:

$$a_1a_2+b_1b_2+c_1c_2 = 0$$

In vector notation: $\mathbf{b_1}\cdot\mathbf{b_2} = 0$.

Lines need not intersect to be perpendicular: Skew lines can be perpendicular — the condition only requires their direction vectors to be perpendicular.

Example: Lines $(x-1)/2 = (y-2)/1 = (z-3)/(-2)$ and $(x-1)/(-1) = (y-3)/2 = (z+1)/1$:

$$(2)(-1)+(1)(2)+(-2)(1) = -2+2-2 = -2 \neq 0$$

Not perpendicular.