3D Geometry

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Q1. What is the distance between the points A(1, 2, 3) and B(4, 5, 6)? (2023)

Solution:

Distance = √[(4-1)² + (5-2)² + (6-3)²] = √[3² + 3² + 3²] = √27 = 3√3.

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Q2. Determine the coordinates of the centroid of the triangle with vertices A(0, 0, 0), B(4, 0, 0), C(0, 3, 0). (2023)

Solution:

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

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Q3. Determine the coordinates of the centroid of the triangle with vertices A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9). (2021)

Solution:

Centroid G = ((1+4+7)/3, (2+5+8)/3, (3+6+9)/3) = (4, 5, 6).

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Q4. Determine the coordinates of the foot of the perpendicular from the point (1, 2, 3) to the plane x + 2y + 3z = 14. (2023)

Solution:

Using the formula for the foot of the perpendicular, we find the coordinates to be (1, 2, 4).

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Q5. What is the volume of a sphere with radius 5 units? (2021)

Solution:

Volume = (4/3)πr³ = (4/3)π(5)³ = (4/3)π(125) = 500/3π cubic units.

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Q6. What is the volume of a cube with side length 3 units? (2023)

Solution:

Volume = side³ = 3³ = 27 cubic units.

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Q7. What is the equation of the plane passing through the point (1, 2, 3) with normal vector (1, -1, 1)? (2023)

Solution:

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

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Q8. Find the area of a triangle with vertices at 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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Q9. 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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Q10. 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.