3D Geometry

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Q1. Find the coordinates of the midpoint of the line segment joining A(2, -1, 3) and B(4, 3, 5). (2022)

Solution:

Midpoint M = ((2+4)/2, (-1+3)/2, (3+5)/2) = (3, 1, 4).

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Q2. Find the area of the triangle formed by the points A(0, 0, 0), B(1, 0, 0), and C(0, 1, 0). (2023)

Solution:

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

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Q3. What is the equation of a plane passing through the point (1, 2, 3) with normal vector (2, -1, 3)? (2021)

Solution:

Equation of the plane: 2(x-1) - 1(y-2) + 3(z-3) = 0 simplifies to 2x - y + 3z = 12.

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Q4. What is the area of the base of a cone with radius 3? (2023)

Solution:

Area of base = πr² = π(3²) = 9π square units.

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Q5. 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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Q6. What is the length of the diagonal of a cuboid with dimensions 2, 3, and 6? (2022)

Solution:

Diagonal = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7.

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

Solution:

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

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Q9. What is the equation of a plane passing through the point (1, 2, 3) with normal vector (1, 1, 1)? (2022)

Solution:

Equation of the plane: 1(x-1) + 1(y-2) + 1(z-3) = 0 => x + y + z = 6.

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Q10. Calculate the distance from the point P(1, 2, 3) to the origin O(0, 0, 0). (2023)

Solution:

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