Distance and direction cosines in 3D: Class 12 practice
Distance between two points in space is √(Δx² + Δy² + Δz²). Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Pythagoras
- Vectors
The idea behind distance and direction cosines in 3d
Distance between two points in space is √(Δx² + Δy² + Δz²). Direction ratios of the line through them are (Δx, Δy, Δz); divide by the length to get direction cosines (l² + m² + n² = 1).
Step-by-step method
- Find Δx, Δy, Δz.
- Distance = √(sum of squares).
- Direction cosines = each Δ ÷ distance.
Worked example 1 (Basic)
Find the distance between (-5, 2, -3) and (1, 0, 0).
- Δ = (6, -2, 3)
- √(36 + 4 + 9) = 7
- Answer: 7
Worked example 2 (Standard)
Find the distance between (-3, -2, -3) and (-2, -6, 5).
- Δ = (1, -4, 8)
- √(1 + 16 + 64) = 9
- Answer: 9
Worked example 3 (Challenge)
Find the direction cosines of the line from (-4, -5, -1) to (0, -9, 6).
- Direction ratios (4, -4, 7), length 9
- l, m, n = 4/9, -4/9, 7/9
- Answer: 4/9, -4/9, 7/9
Common mistakes and misconceptions
- Not squaring negative differences
- Using direction ratios as cosines
Practice questions
- (Basic) Find the distance between (-4, -5, -5) and (2, -7, -2).
- (Basic) Find the distance between (-3, -4, 2) and (-1, -7, 8).
- (Standard) Find the distance between (0, 2, -5) and (2, -1, 1).
- (Standard) Find the distance between (-4, -2, -2) and (-3, -4, 0).
- (Challenge) Find the direction cosines of the line from (-1, 0, 2) to (1, -3, 8).
- (Challenge) Find the direction cosines of the line from (-4, 5, 3) to (-3, 1, 11).
Answer key with full working
- Q1: Δ = (6, -2, 3) → √(36 + 4 + 9) = 7 → Answer: 7
- Q2: Δ = (2, -3, 6) → √(4 + 9 + 36) = 7 → Answer: 7
- Q3: Δ = (2, -3, 6) → √(4 + 9 + 36) = 7 → Answer: 7
- Q4: Δ = (1, -2, 2) → √(1 + 4 + 4) = 3 → Answer: 3
- Q5: Direction ratios (2, -3, 6), length 7 → l, m, n = 2/7, -3/7, 6/7 → Answer: 2/7, -3/7, 6/7
- Q6: Direction ratios (1, -4, 8), length 9 → l, m, n = 1/9, -4/9, 8/9 → Answer: 1/9, -4/9, 8/9
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 distance and direction cosines in 3d 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.