The third dimension brings depth to mathematics, literally and figuratively. As we venture into the world of 3D Geometry, we find ourselves navigating through planes, lines, and points in space, observing their relationships and intricacies. This field has profound implications, from architectural designs to video game graphics. To ensure you’re on the right track in understanding this pivotal topic, we’ve curated 20 multiple-choice questions on Three-Dimensional Geometry, each followed by its answer and a brief explanation.
1. Which of the following is the direction ratio of the line parallel to the x-axis?
Answer:
Explanation:
A line parallel to the x-axis will have a direction ratio along the x-axis.
2. The distance between the points (1, 2, 3) and (1, 2, 5) is:
Answer:
Explanation:
Using the distance formula, the only difference is in the z-coordinate. Hence, the distance is |5-3| = 2 units.
3. Two lines are parallel if:
Answer:
Explanation:
Parallel lines have the same direction. Thus, both their direction ratios are proportional, and their direction cosines are equal.
4. The angle between the lines with direction cosines l = 1/√3, m = 1/√3, and n = 1/√3 is:
Answer:
Explanation:
The angle between the lines is given by cos⁻¹(l), which equals 60° for the given direction cosines.
5. The line passing through the points (1, 2, 3) and (4, 5, 6) has the direction ratios:
Answer:
Explanation:
Direction ratios are the difference in coordinates, (4-1, 5-2, 6-3) = (3, 3, 3).
6. If a line has the equation r = a + λb, then:
Answer:
Explanation:
In the vector equation of a line, 'a' is a fixed point, 'b' gives direction, and λ is a scalar parameter.
7. The equation of the XY plane is:
Answer:
Explanation:
The XY plane is parallel to both the X and Y axes and perpendicular to the Z axis. Hence, z = 0.
8. Which of the following points does not lie in the plane x + y + z = 6?
Answer:
Explanation:
Plugging (2, 2, 3) into the equation gives x + y + z = 7, which is not equal to 6.
9. The line parallel to the Z-axis and passing through the point (1, 2, 3) has the equation:
Answer:
Explanation:
A line parallel to the Z-axis will have fixed x and y coordinates.
10. Two lines are coplanar if:
Answer:
Explanation:
If two lines are either parallel or intersecting, they lie in the same plane and are coplanar.
11. The direction cosines of a line perpendicular to the line with direction cosines (1/√2, 1/√2, 0) are:
Answer:
Explanation:
The line perpendicular to a given line in the XY plane will be along the Z-axis, with direction cosines (0, 0, 1).
12. The point of intersection of the lines r = a1 + λb1 and r = a2 + μb2 is:
Answer:
Explanation:
The point of intersection depends on the values of λ and μ. If the lines intersect, there will be a unique solution for λ and μ.
13. If the direction cosines of a line are (l, m, n), then:
Answer:
Explanation:
This is the fundamental relation between direction cosines.
14. The line with direction ratios 2, 2, 2 will be parallel to the line with direction ratios:
Answer:
Explanation:
Two lines are parallel if their direction ratios are proportional.
15. The plane parallel to the XZ plane and at a distance of 3 units from the origin is:
Answer:
Explanation:
A plane parallel to the XZ plane will have a fixed y-coordinate.
16. If the angle between two lines is 90°, then:
Answer:
Explanation:
Two lines are perpendicular if the dot product of their direction ratios is zero.
17. The line r = a + λb is parallel to:
Answer:
Explanation:
The vector 'b' gives the direction of the line in its vector equation.
18. The foot of the perpendicular from the point (1, 2, 3) on the X-axis is:
Answer:
Explanation:
The foot of the perpendicular on the X-axis from any point will have its x-coordinate unchanged and y, z-coordinates as zero.
19. The angle between the planes 2x – y + 2z = 5 and x + y – z = 4 is:
Answer:
Explanation:
Using the formula for the angle between two planes, and taking the dot product of their normal vectors.
20. If two planes are parallel, then:
Answer:
Explanation:
Parallel planes have parallel normal vectors.