Three Dimensional Geometry

Topics in this chapter

• 3D Coordinates and Vectors
• Equations of Lines in 3D
• Equations of Planes in 3D
• Angles Between Lines and Planes
• Intersections of Lines and Planes
• Distances and Perpendiculars in 3D

Cheat sheet

Practice questions

1. Question 1Q-95E828

   The direction cosines of a line whose direction ratios are 2, -1 and 2 are:

   1. A.(2/3, -1/3, 2/3)
   2. B.(2, -1, 2)
   3. C.(1/2, -1/2, 1/2)
   4. D.(2/√5, -1/√5, 2/√5)
   See the answer and solution
2. Question 2Q-5BB42A

   The shortest distance between the skew lines r = (1, 0, 1) + λ(1, 1, 0) and r = (0, 1, 0) + μ(1, 0, 1) is:

   1. A.1/√3
   2. B.√3
   3. C.1
   4. D.2/√3
   See the answer and solution
3. Question 3Q-81C23A

   Find the equation of the plane passing through the point (1, 2, -1) and perpendicular to the line whose direction ratios are 2, -1 and 2.

   See the answer and solution
4. Question 4Q-C8539B

   The vector equation of the line passing through the points A(1, -2, 3) and B(3, 1, -1) is:

   1. A.r = (i - 2j + 3k) + λ(2i + 3j - 4k)
   2. B.r = (i - 2j + 3k) + λ(3i + 2j - 4k)
   3. C.r = (i + 2j + 3k) + λ(2i + 3j + 4k)
   4. D.r = (3i + j - k) + λ(i - 2j + 3k)
   See the answer and solution
5. Question 5Q-BF04BB

   The acute angle between the planes 2x - y + 2z = 5 and x + 2y + 2z = 7 has cosine equal to:

   1. A.1/9
   2. B.2/9
   3. C.4/9
   4. D.8/9
   See the answer and solution