3D Geometry

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Q1. Find the distance between the parallel planes 2x + 3y + 4z = 5 and 2x + 3y + 4z = 10. (2023)

Solution:

Distance = |d1 - d2| / √(a² + b² + c²) = |5 - 10| / √(2² + 3² + 4²) = 5 / √29.

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Q2. Calculate the distance from the point (1, 2, 3) to the origin (0, 0, 0). (2021)

Solution:

Distance = √[(1-0)² + (2-0)² + (3-0)²] = √[1 + 4 + 9] = √14.

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Q3. Determine the coordinates of the centroid of the triangle with vertices A(0, 0, 0), B(0, 4, 0), and C(3, 0, 0). (2021)

Solution:

Centroid = ((0+0+3)/3, (0+4+0)/3, (0+0+0)/3) = (1, 1.33, 0).

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Q4. What is the volume of a cylinder with radius 2 and height 5? (2023)

Solution:

Volume = πr²h = π(2²)(5) = 20π cubic units.

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Q5. What is the area of a triangle with vertices at A(0, 0, 0), B(1, 0, 0), and C(0, 1, 0)? (2022)

Solution:

Area = 0.5 * base * height = 0.5 * 1 * 1 = 0.5 square units.

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Q6. Determine the coordinates of the centroid of the triangle with vertices A(0, 0, 0), B(6, 0, 0), and C(0, 8, 0). (2023)

Solution:

Centroid = ((0+6+0)/3, (0+0+8)/3, (0+0+0)/3) = (2, 2.67, 0).

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Q7. What is the volume of a cube with side length 3 units? (2023)

Solution:

Volume = side³ = 3³ = 27 cubic units.

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Q8. What is the distance between the parallel planes x + 2y + 3z = 4 and x + 2y + 3z = 10? (2023)

Solution:

Distance = |d1 - d2| / √(a² + b² + c²) = |4 - 10| / √(1² + 2² + 3²) = 6 / √14.

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Q9. Find the length of the diagonal of a cuboid with dimensions 3, 4, and 12 units. (2020)

Solution:

Diagonal = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13.

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Q10. Find the area of the triangle formed by the points A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9). (2022)

Solution:

The points are collinear, hence the area = 0.