Angle between two lines in space: Class 12 practice
For lines with direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂), cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / (√(a₁² + b₁² + c₁²) √(a₂² + b₂² + c₂²)). Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Dot product
The idea behind angle between two lines in space
For lines with direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂), cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / (√(a₁² + b₁² + c₁²) √(a₂² + b₂² + c₂²)). They are perpendicular when the numerator is 0 and parallel when the ratios are proportional.
Step-by-step method
- Write both direction ratios.
- Compute the dot product and both magnitudes.
- θ = cos⁻¹(…).
Worked example 1 (Basic)
Find the angle between lines with direction ratios (-1, -1, 4) and (3, 3, -12).
- Dot = -54; |u| = 4.243, |v| = 12.728
- θ = 0°
- Answer: 0°
Worked example 2 (Standard)
Find the angle between lines with direction ratios (-2, -1, 4) and (-1, 2, 0).
- Dot = 0; |u| = 4.583, |v| = 2.236
- θ = 90°
- Answer: 90°
Worked example 3 (Challenge)
Find the angle between lines with direction ratios (-4, 2, 2) and (-2, -3, 3).
- Dot = 8; |u| = 4.899, |v| = 4.69
- θ = 69.63°
- Answer: 69.63°
Common mistakes and misconceptions
- Using points instead of direction ratios
- Forgetting the absolute value
Practice questions
- (Basic) Find the angle between lines with direction ratios (2, -1, 1) and (-4, 2, -2).
- (Basic) Find the angle between lines with direction ratios (3, 3, 4) and (-9, -9, -12).
- (Standard) Find the angle between lines with direction ratios (-3, 0, 1) and (0, 3, 0).
- (Standard) Find the angle between lines with direction ratios (-3, 1, 4) and (1, 3, 0).
- (Challenge) Find the angle between lines with direction ratios (3, -3, 4) and (0, -2, -3).
- (Challenge) Find the angle between lines with direction ratios (2, 1, 2) and (2, -1, -1).
Answer key with full working
- Q1: Dot = -12; |u| = 2.449, |v| = 4.899 → θ = 0° → Answer: 0°
- Q2: Dot = -102; |u| = 5.831, |v| = 17.493 → θ = 0° → Answer: 0°
- Q3: Dot = 0; |u| = 3.162, |v| = 3 → θ = 90° → Answer: 90°
- Q4: Dot = 0; |u| = 5.099, |v| = 3.162 → θ = 90° → Answer: 90°
- Q5: Dot = -6; |u| = 5.831, |v| = 3.606 → θ = 73.42° → Answer: 73.42°
- Q6: Dot = 1; |u| = 3, |v| = 2.449 → θ = 82.18° → Answer: 82.18°
Syllabus note
This skill is listed under "Three Dimensional Geometry" for Class 12 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is angle between two lines in space in?
"Three Dimensional Geometry", Class 12 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.