introduction to 3 dimensional geometry :
Three-dimensional (3D) geometry studies points, lines, and planes in space using three mutually perpendicular axes: the x-axis, y-axis, and z-axis. Each point in 3D space is represented by coordinates (x, y, z), where x is the abscissa (distance along x-axis), y is the ordinate (distance along y-axis), and z is the applicate (distance along z-axis). The three coordinate planes formed by pairs of axes divide the space into eight parts called octants. Important concepts include distance formula, midpoint formula, direction cosines, direction ratios, and equations of lines and planes [1][2][3].
20 MCQs with Explained Answers:
1. Q: What are the coordinates of a point in 3D geometry?
A) (x, y)
B) (x, y, z)
C) (y, z)
D) (x, z)
Answer: B
Explanation: In 3D geometry, a point is represented as (x, y, z)[1].
2. Q: How many octants are formed by the three coordinate planes?
A) 4
B) 6
C) 8
D) 12
Answer: C
Explanation: Three mutually perpendicular planes divide 3D space into 8 octants[2].
3. Q: The point (0, 0, 3) lies on the?
A) X-axis
B) Y-axis
C) Z-axis
D) XY-plane
Answer: C
Explanation: Points with zero x and y and non-zero z lie on the z-axis[10].
4. Q: The distance between points (x1, y1, z1) and (x2, y2, z2) is given by?
A) √[(x2-x1)² + (y2-y1)²]
B) √[(x2-x1)² + (y2-y1)² + (z2-z1)²]
C) |x2-x1| + |y2-y1| + |z2-z1|
D) None
Answer: B
Explanation: The 3D distance formula includes all three coordinate differences[1].
5. Q: The point (3, 0, 4) lies on?
A) XY-plane
B) YZ-plane
C) XZ-plane
D) X-axis
Answer: C
Explanation: A point with y=0 lies on XZ-plane[10].
6. Q: What is the equation of XY-plane?
A) z=0
B) y=0
C) x=0
D) x+y+z=0
Answer: A
Explanation: XY-plane has z=0[3].
7. Q: The coordinate axes are?
A) Parallel lines
B) Mutually perpendicular lines
C) Intersecting at right angles but non-perpendicular
D) None
Answer: B
Explanation: The x, y, z axes are mutually perpendicular[1].
8. Q: Which of these represents a line in 3D geometry?
A) (x-1)/2 = (y-2)/3 = (z-3)/4
B) x+y+z=1
C) x² + y² + z² = 1
D) None
Answer: A
Explanation: The symmetric form represents lines in 3D[3].
9. Q: Direction cosines are?
A) Cosines of angles a line makes with coordinate axes
B) Ratios of distances
C) Coordinates of midpoints
D) None
Answer: A
Explanation: Direction cosines relate to angles with axes[1].
10. Q: The midpoint between two points in 3D (x1, y1, z1) and (x2, y2, z2) is given by?
A) ((x1+x2)/2, (y1+y2)/2)
B) ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2)
C) (x1-x2, y1-y2, z1-z2)
D) None
Answer: B
Explanation: Midpoint considers all three coordinates[1].
11. Q: The point (-5, 4, 3) lies in which octant?
A) I
B) II
C) III
D) IV
Answer: B
Explanation: Octant determination depends on sign of coordinates[2].
12. Q: If the vector components are (3, -2, 6), what is the magnitude of the vector?
A) √49
B) 7
C) Both A and B
D) None
Answer: C
Explanation: Magnitude = √(3²+(-2)²+6²) = √49 = 7[1].
13. Q: The equation 3x - 4y = 0 represents?
A) Plane containing Y-axis
B) Plane containing X-axis
C) Plane containing Z-axis
D) None
Answer: A
Explanation: Plane contains Y-axis as y is unrestricted[4].
14. Q: The sum of direction cosines of a line is zero. If l, m are two cosines, what is n?
A) l + m
B) - (l + m)
C) lm
D) None
Answer: B
Explanation: Since sum is 0, n = -(l + m)[8].
15. Q: The 3D coordinate system origin is?
A) (0, 0)
B) (0, 0, 0)
C) (1, 1, 1)
D) None
Answer: B
Explanation: Origin in 3D has all coordinates zero[1].
16. Q: The equation x² - x - 2 = 0 in 3D space represents?
A) Pair of parallel planes
B) Pair of lines
C) Perpendicular planes
D) None
Answer: A
Explanation: This equation breaks into factors representing parallel planes[6].
17. Q: The plane equation passing through point (x1, y1, z1) and parallel to XY-plane is?
A) z = z1
B) x = x1
C) y = y1
D) None
Answer: A
Explanation: z coordinate is constant parallel to XY-plane[3].
18. Q: In 3D, the shortest distance is measured along?
A) x-y plane
B) Line segment connecting points
C) xy-axis
D) None
Answer: B
Explanation: Distance between two points is shortest straight line between them[1].
19. Q: Direction ratio defines?
A) Length of line
B) Proportional components of direction vector
C) Distance from origin
D) None
Answer: B
Explanation: Direction ratios indicate orientation of line[1].
20. Q: The point dividing the line segment joining A and B in ratio m:n is given by?
A) ((mx1+nx2)/(m+n), (my1+ny2)/(m+n), (mz1+nz2)/(m+n))
B) ((x1+x2)/2, (y1+y2)/2)
C) (m+n, m-n)
D) None
Answer: A
Explanation: Section formula in 3D considers all three coordinates[1].
21 to 40
Here are 20 more multiple choice questions (21 to 40) on three-dimensional geometry with explained answers:
21. The equation of the plane through the intersection of the planes $$ x + 2y + 3z = 4 $$ and $$ 2x - y + z = 5 $$, and perpendicular to the plane $$ 5x + 3y - 2z = 5 $$ is:
A) $$ 2x + 17y + 7z = 12 $$
B) $$ 17x + 2y + 7z = 12 $$
C) $$ 7x + 2y + 17z = 12 $$
D) None of these
Answer: A
Explanation: The plane must be a linear combination of the first two and orthogonal to the third, solving the system gives the answer [1].
22. The perpendicular distance of the point (3, -4, -5) from the line $$\frac{x-2}{4} = \frac{y+6}{5} = \frac{z-5}{-3}$$ is:
A) 5
B) 6
C) 7
D) 8
Answer: A
Explanation: Use formula for distance from point to line in 3D [2].
23. The vector form of the line passing through points (2, 2, -3) and (1, 3, 5) is:
A) $$ \mathbf{r} = (2,2,-3) + \lambda(-1,1,8) $$
B) $$ \mathbf{r} = (1,3,5) + \lambda(1,-1,-8) $$
C) $$ \mathbf{r} = (2,2,-3) + \lambda(1,1,8) $$
D) None
Answer: A
Explanation: Direction vector = difference of points, substituted in vector form [2].
24. The equation of the plane passing through points P(1, 1, 2) and Q(2, 2, 2) and perpendicular to plane $$6x + 2y + 2z = 3$$ is:
A) $$x + y = 0$$
B) $$x - y = 0$$
C) $$x + y + z = 0$$
D) None
Answer: B
Explanation: Normal to plane is parallel to the given plane’s normal, use conditions accordingly [1].
25. The direction cosines $$l, m, n$$ satisfy the equation $$l + m + n = 0$$. If $$l, m$$ are known, $$n$$ is:
A) $$l + m$$
B) $$- (l + m)$$
C) $$lm$$
D) None
Answer: B
Explanation: By rearrangement of the equation [1].
26. A line has components (3, -2, 6). Its magnitude is:
A) 7
B) $$\sqrt{49}$$
C) Both A and B
D) None
Answer: C
Explanation: Magnitude $$= \sqrt{3^2 + (-2)^2 + 6^2} = 7$$ [1].
27. The angle between lines with direction cosines $$l, m, n$$ and $$m-n, n-l, l-m$$ is:
A) $$90^\circ$$
B) $$120^\circ$$
C) $$60^\circ$$
D) None
Answer: B
Explanation: Using scalar product formula and given cosines sum relations [3].
28. The length of line whose projections on the axes are 9, 12, and 8 is:
A) 15
B) 13
C) 14
D) None
Answer: A
Explanation: Length $$= \sqrt{9^2 + 12^2 + 8^2} = 15$$ [2].
29. The shortest distance between skew diagonals of a cube of edge $$a$$ is:
A) $$a\sqrt{2}$$
B) $$a/ \sqrt{2}$$
C) $$a$$
D) None
Answer: B
Explanation: Using vector projection and cube diagonal properties [2].
30. The distance of a point (2, -3, 6) from the plane $$3x - 6y + 2z + 10 = 0$$ is:
A) 7
B) 9
C) 10
D) None
Answer: B
Explanation: Use distance formula from point to plane [2].
31. The line passing through points (2, -3, 1) and (3, -4, -5) intersects the ZX-plane at:
A) (x, 0, z)
B) (x, y, 0)
C) (0, y, z)
D) None
Answer: A
Explanation: Since ZX-plane means $$y=0$$, find intersection [2].
32. The equation of the plane passing through (3, 1, 2) and perpendicular to vector from A(3, 1, 2) to B(1, -2, -4) is:
A) $$x - 3y + 2z = 1$$
B) $$2x + 3y - z = 4$$
C) $$x + y - z = 0$$
D) None
Answer: B
Explanation: Direction vector used as normal to plane [2].
33. The image of point (1, 6, 3) with respect to the line $$\frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}$$ is:
A) (3, 8, 5)
B) (1, 6, 3)
C) (2, 4, 1)
D) None
Answer: A
Explanation: Use reflection formula with line parametrics [2].
34. The distance of point P(4, 3, 5) from the Y-axis is:
A) $$\sqrt{41}$$
B) 4
C) 5
D) None
Answer: A
Explanation: Distance from Y-axis is $$\sqrt{x^2 + z^2} = \sqrt{4^2 + 5^2} = \sqrt{41}$$ [2].
35. The direction ratios of a line satisfy:
A) $$al + bm + cn = 0$$
B) $$l^2 + m^2 + n^2 = 1$$
C) $$l + m + n = 0$$
D) None
Answer: A
Explanation: General relation for direction ratios [2].
36. The distance between planes $$2x - y + 2z = 1$$ and $$4x - 2y + 4z = 1$$ is:
A) $$\frac{1}{\sqrt{9}}$$
B) Zero
C) $$\frac{1}{2}$$
D) None
Answer: B
Explanation: Planes are parallel scaling corresponds to the same plane, distance zero [2].
37. The line passing through (2, -3, 5) and making equal angles with axes has direction cosines:
A) $$ \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} $$
B) (1, 0, 0)
C) (0, 1, 0)
D) None
Answer: A
Explanation: Equal angles leads to equal direction cosines [2].
38. The vector equation of the line $$\frac{3-x}{3} = \frac{2y-3}{5} = \frac{z}{2}$$ is:
A) $$\mathbf{r} = (3, \frac{3}{2}, 0) + \lambda (-3, 5, 2)$$
B) $$\mathbf{r} = (-3, 5, 2) + \lambda (3, -\frac{3}{2}, 0)$$
C) None
Answer: A
Explanation: Rearranged symmetric form to vector form [2].
39. The equation of the plane whose intercepts are 4, -6, 3 on axes is:
A) $$\frac{x}{4} - \frac{y}{6} + \frac{z}{3} = 1$$
B) $$\frac{x}{4} + \frac{y}{-6} + \frac{z}{3} = 1$$
C) None
Answer: B
Explanation: Intercept form of plane equation [2].
40. The direction of line passing through (-1, 2, 4) and parallel to plane $$3x - 4y + 7z = 2$$ is:
A) Any vector perpendicular to $$ (3, -4, 7) $$
B) $$ (3, -4, 7) $$
C) (1, 1, 1)
D) None
Answer: A
Explanation: Direction vector must be orthogonal to normal of plane [2].
