Three Dimensional Geometry Important Questions.

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1. Q1Medium

   If a line makes angles of 60°, 45°, and 135° with the x, y, and z-axes respectively, what are its direction cosines?

   • A.±(1/2), ±(1/√2), ±(1/2)
   • B.±(1/√2), ±(1/2), ±(1/√2)
   • C.±(1/2), ±(1/√3), ±(1/√2)
   • D.±(1/√3), ±(1/2), ±(1/√2)
   Answer: ±(1/√2), ±(1/2), ±(1/√2)

   Explanation: Direction cosines are the cosines of the angles a line makes with the coordinate axes. Using the given angles, the direction cosines are calculated as cos(60°), cos(45°), and cos(135°), which are ±(1/2), ±(1/√2), and ±(1/√2) respectively. The sign depends on the direction of the line.

2. Q2Medium

   What are the direction ratios of a line with direction cosines (1/2, -1/2, -1/2)?

   • A.1, 1, -1
   • B.1, -1, -1
   • C.-1, 1, -1
   • D.-1, -1, 1
   Answer: 1, -1, -1

   Explanation: Direction ratios are proportional to direction cosines. Multiplying the direction cosines (1/2, -1/2, -1/2) by 2 gives the direction ratios (1, -1, -1). This maintains proportionality while using integer values.

3. Q3Medium

   Which of the following sets of points are collinear? (Points: A(2, 3, 4), B(-1, -2, 1), C(5, 8, 7))

   • A.Collinear
   • B.Not Collinear
   • C.Coplanar but not Collinear
   • D.Cannot be determined
   Answer: Collinear

   Explanation: To check for collinearity, verify if the direction ratios of AB and BC are proportional. AB has direction ratios (-3, -5, -3) and BC has direction ratios (6, 10, 6). These are proportional with a factor of -2, confirming collinearity.

4. Q4Medium

   What is the Cartesian equation of a line passing through point (2, 3, 4) and parallel to the vector (1, 2, 3)?

   • A.(x-2)/1 = (y-3)/2 = (z-4)/-3
   • B.(x-2)/-1 = (y-3)/2 = (z-4)/3
   • C.(x-2)/1 = (y-3)/-2 = (z-4)/3
   • D.(x-2)/1 = (y-3)/2 = (z-4)/3
   Answer: (x-2)/1 = (y-3)/2 = (z-4)/3

   Explanation: The Cartesian equation of a line passing through a point (x₁, y₁, z₁) and parallel to a direction vector (a, b, c) is given by (x-x₁)/a = (y-y₁)/b = (z-z₁)/c. Substituting the values gives (x-2)/1 = (y-3)/2 = (z-4)/3.

5. Q5Medium

   The shortest distance between two skew lines given by r = i + 2j + 3k + λ(i - j + k) and r = 2i - j + k + μ(2i + j - k) is:

   • A.6/√3
   • B.6/√6
   • C.6/√2
   • D.6/√5
   Answer: 6/√6

   Explanation: The shortest distance between skew lines r = a₁ + λb₁ and r = a₂ + μb₂ is |(a₂ - a₁) · (b₁ × b₂)| / |b₁ × b₂|. Here, a₁ = i + 2j + 3k, b₁ = i - j + k, a₂ = 2i - j + k, b₂ = 2i + j - k. Calculating the cross product and dot product yields the distance as 6/√6.

6. Q6Medium

   What is the angle between the lines with direction ratios (1, 1, 0) and (1, 0, 1)?

   • A.cos⁻¹(1/2)
   • B.cos⁻¹(1/√2)
   • C.cos⁻¹(1/√3)
   • D.cos⁻¹(1/3)
   Answer: cos⁻¹(1/√3)

   Explanation: The angle θ between two lines with direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂) is given by cosθ = (a₁a₂ + b₁b₂ + c₁c₂) / √(a₁² + b₁² + c₁²)√(a₂² + b₂² + c₂²). Substituting the values gives cosθ = (1*1 + 1*0 + 0*1) / √(1² + 1² + 0²)√(1² + 0² + 1²) = 1/√3.

7. Q7Medium

   Which of the following is the vector equation of a line passing through point (1, 2, 3) and parallel to the vector (2, 3, 4)?

   • A.r = (1, 2, 3) + λ(-2, -3, -4)
   • B.r = (1, 2, 3) + λ(2, -3, 4)
   • C.r = (1, 2, 3) + λ(2, 3, 4)
   • D.r = (1, -2, 3) + λ(2, 3, 4)
   Answer: r = (1, 2, 3) + λ(2, 3, 4)

   Explanation: The vector equation of a line passing through a point a = (x₁, y₁, z₁) and parallel to a vector b = (a, b, c) is r = a + λb. Substituting the given values gives r = (1, 2, 3) + λ(2, 3, 4).

8. Q8Medium

   If the direction ratios of a line are (2, -2, 1), what are its direction cosines?

   • A.(2/√5, -2/√5, 1/√5)
   • B.(2/3, -2/3, 1/3)
   • C.(2/4, -2/4, 1/4)
   • D.(2/√10, -2/√10, 1/√10)
   Answer: (2/3, -2/3, 1/3)

   Explanation: The direction cosines are the normalized direction ratios. The magnitude of the direction ratios (2, -2, 1) is √(2² + (-2)² + 1²) = 3. Normalizing gives direction cosines (2/3, -2/3, 1/3).

9. Q9Medium

   What is the shortest distance between the parallel lines r = (i + j + k) + λ(2i - j + k) and r = (2i - j + 3k) + μ(2i - j + k)?

   • A.1/√3
   • B.1/√6
   • C.1/√2
   • D.1/√5
   Answer: 1/√6

   Explanation: The shortest distance between parallel lines r = a₁ + λb and r = a₂ + μb is |(a₂ - a₁) × b| / |b|. Here, a₁ = i + j + k, a₂ = 2i - j + 3k, b = 2i - j + k. Calculating the cross product and magnitude yields the distance as 1/√6.

10. Q10Medium

    Which of the following is the Cartesian equation of the line that passes through (1, 2, 3) and has direction ratios (1, 2, 3)?

    • A.(x-1)/1 = (y-2)/-2 = (z-3)/3
    • B.(x-1)/-1 = (y-2)/2 = (z-3)/3
    • C.(x-1)/1 = (y-2)/2 = (z-3)/-3
    • D.(x-1)/1 = (y-2)/2 = (z-3)/3
    Answer: (x-1)/1 = (y-2)/2 = (z-3)/3

    Explanation: The Cartesian equation of a line passing through a point (x₁, y₁, z₁) and having direction ratios (a, b, c) is given by (x-x₁)/a = (y-y₁)/b = (z-z₁)/c. Substituting the values gives (x-1)/1 = (y-2)/2 = (z-3)/3.

11. Q11Medium

    How do you determine if three points A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9) are collinear?

    • A.They are collinear
    • B.They are not collinear
    • C.They are coplanar but not collinear
    • D.Cannot be determined
    Answer: They are collinear

    Explanation: To check collinearity, verify if the direction ratios of AB and BC are proportional. AB has direction ratios (3, 3, 3) and BC has direction ratios (3, 3, 3). Since these are identical, the points are collinear.

12. Q12Medium

    What is the angle between the lines with direction cosines (1/√3, 1/√3, 1/√3) and (1/2, 1/2, 0)?

    • A.cos⁻¹(1/3)
    • B.cos⁻¹(1/√2)
    • C.cos⁻¹(1/2)
    • D.cos⁻¹(√2/2)
    Answer: cos⁻¹(1/√2)

    Explanation: The angle θ between two lines with direction cosines (l₁, m₁, n₁) and (l₂, m₂, n₂) is given by cosθ = l₁l₂ + m₁m₂ + n₁n₂. Substituting the values gives cosθ = (1/√3)(1/2) + (1/√3)(1/2) + (1/√3)(0) = 1/√2.

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