JEE (Main) Mathematics - Three Dimensional Geometry: Section Formula, Direction Ratios, Direction Cosines, and Angle Between Two Intersecting Lines

16. The section formula in 3D coordinates divides a line segment joining two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) in the ratio m:n. The coordinates of the point dividing the line segment are:
- A) (mx₂ + nx₁) / (m + n), (my₂ + ny₁) / (m + n), (mz₂ + nz₁) / (m + n)
- B) (mx₁ + nx₂) / (m + n), (my₁ + ny₂) / (m + n), (mz₁ + nz₂) / (m + n)
- C) (mx₂ + ny₁) / (m + n), (my₁ + nz₂) / (m + n), (mz₁ + nx₂) / (m + n)
- D) None of the above
Answer: B) (mx₁ + nx₂) / (m + n), (my₁ + ny₂) / (m + n), (mz₁ + nz₂) / (m + n)
17. If a point P divides the line segment joining points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) in the ratio 3:2, then the coordinates of P are:
- A) (3x₁ + 2x₂) / 5, (3y₁ + 2y₂) / 5, (3z₁ + 2z₂) / 5
- B) (3x₂ + 2x₁) / 5, (3y₂ + 2y₁) / 5, (3z₂ + 2z₁) / 5
- C) (3x₁ + 2x₂) / 3, (3y₁ + 2y₂) / 3, (3z₁ + 2z₂) / 3
- D) None of the above
Answer: A) (3x₁ + 2x₂) / 5, (3y₁ + 2y₂) / 5, (3z₁ + 2z₂) / 5
18. The direction cosines of a line are:
- A) The cosines of the angles that the line makes with the coordinate axes
- B) The sine of the angles that the line makes with the coordinate axes
- C) The tangents of the angles that the line makes with the coordinate axes
- D) None of the above
Answer: A) The cosines of the angles that the line makes with the coordinate axes
19. The direction ratios of a line are:
- A) The components of the direction cosines
- B) The coordinates of a point on the line
- C) The cosines of the angles that the line makes with the coordinate axes
- D) Proportional to the direction cosines of the line
Answer: D) Proportional to the direction cosines of the line
20. If a line has direction cosines (l, m, n), then the condition for these to be valid is:
- A) l² + m² + n² = 0
- B) l² + m² + n² = 1
- C) l² + m² + n² = 2
- D) None of the above
Answer: B) l² + m² + n² = 1
21. The angle between two intersecting lines with direction ratios (l₁, m₁, n₁) and (l₂, m₂, n₂) is given by:
- A) cos θ = (l₁ l₂ + m₁ m₂ + n₁ n₂) / √(l₁² + m₁² + n₁²) √(l₂² + m₂² + n₂²)
- B) cos θ = (l₁ l₂ + m₁ m₂) / √(l₁² + m₁²) √(l₂² + m₂²)
- C) cos θ = (l₁ + l₂) / (m₁ + m₂)
- D) cos θ = (l₁ + m₁) / (n₁ + l₂)
Answer: A) cos θ = (l₁ l₂ + m₁ m₂ + n₁ n₂) / √(l₁² + m₁² + n₁²) √(l₂² + m₂² + n₂²)
22. If the direction cosines of a line are given by (1/√3, 1/√3, 1/√3), then the direction ratios of the line are:
- A) (1, 1, 1)
- B) (2, 2, 2)
- C) (3, 3, 3)
- D) (√3, √3, √3)
Answer: A) (1, 1, 1)
23. The angle between two intersecting lines having direction ratios (l₁, m₁, n₁) and (l₂, m₂, n₂) is 90°. What is the relation between their direction ratios?
- A) l₁ l₂ + m₁ m₂ + n₁ n₂ = 1
- B) l₁ l₂ + m₁ m₂ + n₁ n₂ = 0
- C) l₁² + m₁² + n₁² = l₂² + m₂² + n₂²
- D) l₁ l₂ + m₁ m₂ + n₁ n₂ = -1
Answer: B) l₁ l₂ + m₁ m₂ + n₁ n₂ = 0
24. The direction ratios of the line joining the points A(1, 2, 3) and B(4, 5, 6) are:
- A) (3, 3, 3)
- B) (2, 3, 1)
- C) (1, 1, 1)
- D) (3, 3, 3)
Answer: A) (3, 3, 3)
25. The direction cosines of a line are (1/2, 1/2, 1/2). The direction ratios of the line are:
- A) (2, 2, 2)
- B) (1, 1, 1)
- C) (0, 0, 0)
- D) (√2, √2, √2)
Answer: A) (2, 2, 2)
26. The section formula is used to find:
- A) The midpoint of two points
- B) The coordinates of a point dividing the line in a given ratio
- C) The distance between two points
- D) None of the above
Answer: B) The coordinates of a point dividing the line in a given ratio
27. For two lines to be perpendicular, the dot product of their direction ratios must be:
- A) 1
- B) -1
- C) 0
- D) None of the above
Answer: C) 0
28. The angle between two lines with direction cosines (l₁, m₁, n₁) and (l₂, m₂, n₂) is:
- A) cos θ = (l₁ l₂ + m₁ m₂ + n₁ n₂) / (l₁² + m₁² + n₁²) (l₂² + m₂² + n₂²)
- B) sin θ = (l₁ l₂ + m₁ m₂ + n₁ n₂) / (l₁² + m₁² + n₁²) (l₂² + m₂² + n₂²)
- C) cos θ = (l₁ l₂ - m₁ m₂ + n₁ n₂)
- D) cos θ = (l₁² + m₁² + n₁²) / (l₂² + m₂² + n₂²)
Answer: A) cos θ = (l₁ l₂ + m₁ m₂ + n₁ n₂) / √(l₁² + m₁² + n₁²) √(l₂² + m₂² + n₂²)

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JEE (Main) Mathematics - Three Dimensional Geometry: Section Formula, Direction Ratios, Direction Cosines, and Angle Between Two Intersecting Lines

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