Introduction to Three Dimensional Geometry — Class 11 Mathematics NCERT Solutions (Free)

Free step-by-step NCERT solutions for Class 11 Mathematics chapter "Introduction to Three Dimensional Geometry" — 8 important questions with detailed answers for CBSE board exam preparation.

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TL;DR: Free step-by-step NCERT solutions for Class 11 Mathematics chapter "Introduction to Three Dimensional Geometry" — 8 important questions with detailed…

Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated Jul 23, 2026

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Key Questions Covered:

Find the distance between the points A(1, 2, 3) and B(4, 5, 6) in 3D space.
Find the coordinates of the point that divides the line segment joining A(2, …
Find the direction cosines of the line joining the points A(2, 3, 1) and B(6,…
Find the angle between the lines whose direction cosines are (1/√3, 1/√3, 1/√…
Find the centroid of the tetrahedron with vertices at A(1, 2, 3), B(4, 5, 6),…
Find the equation of the line passing through the point (1, 2, 3) and having …
+ 2 more questions in the full chapter

Solutions Summary:

Question	Status
Find the distance between the points A(1, 2, 3) and B(4, …	✓ Solved
Find the coordinates of the point that divides the line s…	✓ Solved
Find the direction cosines of the line joining the points…	✓ Solved
Find the angle between the lines whose direction cosines …	✓ Solved
Find the centroid of the tetrahedron with vertices at A(1…	✓ Solved
Find the equation of the line passing through the point (…	✓ Solved

Showing 6 of 8 questions

Q1: Find the distance between the points A(1, 2, 3) and B(4, 5, 6) in 3D space.

Given points: A(1, 2, 3) and B(4, 5, 6) Step 1: Use the distance formula in 3D: d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²] Step 2: Substitute the coordinates: d = √[(4 - 1)² + (5 - 2)² + (6 - 3)²] d = √[3² + 3² + 3²] d = √[9 + 9 + 9] d = √27 d = 3√3 Step 3: Simplify: d = 3√3 ≈ 5.196 units Final Answer: Distance AB = 3√3 units

Q2: Find the coordinates of the point that divides the line segment joining A(2, 3, 4) and B(8, 9, 10) in the ratio 1:2 internally.

Given: A(2, 3, 4), B(8, 9, 10), and ratio m:n = 1:2 Step 1: Use the section formula for internal division: P = [(m × x₂ + n × x₁)/(m + n), (m × y₂ + n × y₁)/(m + n), (m × z₂ + n × z₁)/(m + n)] Step 2: Substitute m = 1, n = 2: P = [(1 × 8 + 2 × 2)/(1 + 2), (1 × 9 + 2 × 3)/(1 + 2), (1 × 10 + 2 × 4)/(1 + 2)] Step 3: Calculate each coordinate: x-coordinate: (8 + 4)/3 = 12/3 = 4 y-coordinate: (9 + 6)/3 = 15/3 = 5 z-coordinate: (10 + 8)/3 = 18/3 = 6 Step 4: Verify: Distance AP = √[(4-2)² + (5-3)² ...

Q3: Find the direction cosines of the line joining the points A(2, 3, 1) and B(6, 9, 7).

Given points: A(2, 3, 1) and B(6, 9, 7) Step 1: Find the direction ratios: Direction ratios = (6 - 2, 9 - 3, 7 - 1) = (4, 6, 6) Step 2: Find the magnitude of direction ratios: |d| = √(4² + 6² + 6²) |d| = √(16 + 36 + 36) |d| = √88 |d| = 2√22 Step 3: Find direction cosines by dividing each ratio by the magnitude: Direction cosines = (4/(2√22), 6/(2√22), 6/(2√22)) = (2/√22, 3/√22, 3/√22) Step 4: Rationalize: = (2√22/22, 3√22/22, 3√22/22) = (√22/11, 3√22/22, 3√22/22) Step 5: Verify that l² + m²...

Q4: Find the angle between the lines whose direction cosines are (1/√3, 1/√3, 1/√3) and (1/√2, 1/√2, 0).

Given direction cosines: Line 1: l₁ = 1/√3, m₁ = 1/√3, n₁ = 1/√3 Line 2: l₂ = 1/√2, m₂ = 1/√2, n₂ = 0 Step 1: Use the formula for angle between lines: cos θ = |l₁l₂ + m₁m₂ + n₁n₂| Step 2: Calculate the dot product: l₁l₂ = (1/√3)(1/√2) = 1/√6 m₁m₂ = (1/√3)(1/√2) = 1/√6 n₁n₂ = (1/√3)(0) = 0 Step 3: Sum the products: l₁l₂ + m₁m₂ + n₁n₂ = 1/√6 + 1/√6 + 0 = 2/√6 = 2√6/6 = √6/3 Step 4: Find the angle: cos θ = |√6/3| = √6/3 θ = cos⁻¹(√6/3) Step 5: Approximate: √6 ≈ 2.449 √6/3 ≈ 0.816 θ ≈ cos⁻¹(0.8...

Q5: Find the centroid of the tetrahedron with vertices at A(1, 2, 3), B(4, 5, 6), C(7, 8, 9), and D(2, 3, 4).

Given vertices: A(1, 2, 3), B(4, 5, 6), C(7, 8, 9), D(2, 3, 4) Step 1: Use the formula for centroid of a tetrahedron: Centroid G = [(x₁ + x₂ + x₃ + x₄)/4, (y₁ + y₂ + y₃ + y₄)/4, (z₁ + z₂ + z₃ + z₄)/4] Step 2: Calculate x-coordinate: x = (1 + 4 + 7 + 2)/4 = 14/4 = 7/2 = 3.5 Step 3: Calculate y-coordinate: y = (2 + 5 + 8 + 3)/4 = 18/4 = 9/2 = 4.5 Step 4: Calculate z-coordinate: z = (3 + 6 + 9 + 4)/4 = 22/4 = 11/2 = 5.5 Step 5: Write the centroid: G = (7/2, 9/2, 11/2) or (3.5, 4.5, 5.5) Final...

Q6: Find the equation of the line passing through the point (1, 2, 3) and having direction ratios 2:3:4.

Given: Point P₀(1, 2, 3) and direction ratios a:b:c = 2:3:4 Step 1: Use the parametric form of a line: x = x₀ + at y = y₀ + bt z = z₀ + ct Where t is a parameter. Step 2: Substitute the values: x = 1 + 2t y = 2 + 3t z = 3 + 4t Step 3: Alternatively, use the symmetric form: (x - x₀)/a = (y - y₀)/b = (z - z₀)/c (x - 1)/2 = (y - 2)/3 = (z - 3)/4 Step 4: Verify by choosing a value of parameter: At t = 0: Point is (1, 2, 3) ✓ At t = 1: Point is (3, 5, 7) Check: (3 - 1)/2 = 1, (5 - 2)/3 = 1, (7 -...

Showing 6 of 8 questions. Visit the full page for complete solutions.