Q.Find the angle between the lines whose direction ratios are $a, b, c$ and $b-c, c-a, a-b$.

Angle Between Two Lines

In space, the angle between two lines is measured through their directions, not their positions — two lines that never meet still have a well-defined angle between them (the angle you would see if you slid one across to meet the other).

So the angle between the lines is just the angle between their direction vectors. If the lines run along $\vec{b_1}$ and $\vec{b_2}$,

Why the absolute value

A line has two opposite directions, so $\vec{b}$ and $-\vec{b}$ describe the same line. The modulus in the numerator picks the acute angle ($0^\circ \le \theta \le 90^\circ$), which is the convention for the angle between lines.

In Cartesian form

If the lines have direction ratios $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$,

If instead you know the direction cosines $(l_1,m_1,n_1)$ and $(l_2,m_2,n_2)$, the denominators are both $1$ and $\cos\theta = |l_1l_2+m_1m_2+n_1n_2|$.

Two special cases

• Parallel: the direction ratios are proportional, $\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}$.
• Perpendicular: the dot product vanishes, $a_1a_2+b_1b_2+c_1c_2 = 0$.

Example

Lines with directions $\vec{b_1}=(1,2,2)$ and $\vec{b_2}=(2,2,1)$:

so $\theta = \cos^{-1}\!\frac{8}{9}$.

Finding the angle between two lines using their direction ratios or direction cosines is one of the most exam-relevant results in the NCERT Class 12 Three Dimensional Geometry chapter, tested in CBSE boards, JEE Main and several state CETs. "Angle between two lines in 3D formula" is a commonly searched revision topic, and the same absolute-value trick reappears later for angles between lines and planes.