Angle between lines
\[\cos\theta=\frac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{\sum a_1^2}\sqrt{\sum a_2^2}}\]
Three Dimensional Geometry
Angle Between Two Lines
Understand
The angle \(\theta\) between two lines is the angle between their direction vectors \(\vec b_1,\vec b_2\) (we take the acute angle):
\[\cos\theta=\frac{|\vec b_1\cdot\vec b_2|}{|\vec b_1||\vec b_2|}=\frac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}.\]
- Perpendicular lines: \(a_1a_2+b_1b_2+c_1c_2=0\).
- Parallel lines: \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\).
Example: lines with direction ratios \(1,2,2\) and \(2,-1,2\): \(\cos\theta=\frac{|2-2+4|}{3\cdot3}=\frac49\).
Key Concepts
- Use direction ratios only (points do not matter).
- Perpendicular ⇔ sum of products of DRs is 0.
- Parallel ⇔ DRs proportional.
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The angle between lines depends only on their directions.
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