Angle between two lines
The angle between two lines is found from their direction vectors. The anchor points do not affect the angle. “The angle between two lines” means the acute angle, or 90° when the lines are perpendicular.
Use directions, not position vectors
- Extract one direction vector from each line equation.
- The outer vertical bars on the numerator mean take the absolute value of the scalar dot product.
- Reversing a line direction changes the sign of the dot product but not the acute angle between the lines.
Common mistake
Do not use the anchor vector in the dot product. Only the two direction vectors control the line angle.
Calculate the ratio before inverse cosine
Worked example
Two line directions
Answer
(Pure Mathematics 2 & 3 Coursebook, Ch. 9.3–9.4)
Repair an obtuse vector angle for two lines
Your turnIndependent check [4 marks]
Find the acute angle between lines with directions and .
Show worked answer
The negative dot product gives the obtuse vector angle. For the acute line angle, use .
Answer