Three Dimensional Geometry — Questions & Answers

Sample questions from Three Dimensional Geometry

1. Q1. Two lines through the origin have direction cosines (1/3, 2/3, 2/3) and (2/3, -2/3, 1/3). The angle between them is:

   • A.0 degrees
   • B.60 degrees
   • C.90 degrees✓
   • D.45 degrees
   Solution

   The dot product is (1/3)(2/3) + (2/3)(-2/3) + (2/3)(1/3) = 2/9 - 4/9 + 2/9 = 0, so the two lines are perpendicular.

2. Q2. The plane passing through the origin and containing the line (x-2)/1 = (y-1)/(-2) = (z+3)/1 is:

   • A.2x + y - 3z = 0
   • B.x - 2y + z = 0
   • C.x - y + z = 0
   • D.x + y + z = 0✓
   Solution

   The plane contains the origin and the point (2,1,-3), so its normal is perpendicular to both (2,1,-3) and the direction (1,-2,1): (2,1,-3) x (1,-2,1) = ((1)(1) - (-3)(-2), (-3)(1) - (2)(1), (2)(-2) - (1)(1)) = (-5, -5, -5), which reduces to (1, 1, 1). The plane is x + y + z = 0.

3. Q3. The distance of the point (1,2,3) from the plane x + y + z = 9, measured along the line through it with direction ratios (2,3,6), is:

   • A.sqrt(3) units
   • B.7/11 units
   • C.21/11 units✓
   • D.3/sqrt(3) units
   Solution

   The line meets the plane where (1 + 2t) + (2 + 3t) + (3 + 6t) = 6 + 11t = 9, so t = 3/11. The measured distance is |t| * |(2,3,6)| = (3/11)(7) = 21/11 units.

4. Q4. The area of the triangle with vertices A(2,1,0), B(3,3,2) and C(1,2,3) is:

   • A.sqrt(59) square units
   • B.sqrt(35)/2 square units
   • C.5*sqrt(2)/2 square units✓
   • D.5*sqrt(2) square units
   Solution

   AB = (1,2,2) and AC = (-1,1,3). AB x AC = ((2)(3) - (2)(1), (2)(-1) - (1)(3), (1)(1) - (2)(-1)) = (4, -5, 3), whose magnitude is sqrt(16 + 25 + 9) = sqrt(50) = 5 sqrt(2). Area = (1/2)(5 sqrt(2)) = 5 sqrt(2)/2 square units.

5. Q5. The image of the point (2,-3,4) in the line x/2 = (y - 4)/(-1) = (z + 4)/2 is:

   • A.(6, 1, 2)
   • B.(8, 4, 1)
   • C.(4, -1, 6)
   • D.(10, 5, 0)✓
   Solution

   The foot of the perpendicular is F(6,1,2), obtained at t = 3. The image is 2F - P = (12 - 2, 2 + 3, 4 - 4) = (10, 5, 0).

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