Introduction to 3D

 

One mark questions:

 1. The three coordinate planes divide the space into how many parts? What are they known as?

 2. What are the coordinates of any point on X-axis?

 3. What are the coordinates of any point on Y-axis?

 4. What are the coordinates of any point on Z-axis?

 5. A point is on the X-axis. What are its y and z coordinates?

 6. A point is in the XZ plane, what can you say about its y coordinate?

 7. Name the plane determined by X and Y axes taken together.

 8. What are the coordinates of any point in the XY-plane?

 9. What are the coordinates of any point in the YZ-plane?

10. What are the coordinates of any point in the XZ-plane?

11. Name the octant in which the point (2,-4,-7) lie.

12.Name the octants in which the following points lie, (5, 2, 3). 

13. Name the octants in which the following points lie, (-5, 4, 3). 

14. Name the octants in which the following points lie, (4,-3, 5). 

15. Name the octants in which the following points lie, (7, 4,-3). 

16. Name the octants in which the following points lie, (-5,-4, 7). 

17. Name the octants in which the following points lie, (-5,-3,-2). 

18. Find the image of (-2, 3, 4) in the yz-plane. 

19. Find the image of (-5, 4,-3) in the xz-plane. 

20. Find the image of (5, 2,-7) in the xy-plane. 

21.Find the image of (-5, 0, 3) in the xz-plane. 

22.Fill in the blanks: 

(i) Z-coordinates of a point lying in XY-plane is _______________ 

(ii) The x-axis and y-axis taken together determine a plane known as ____________

(iii) The coordinates of point on x-axis of the form ________________

12. Find the distance of the point (3, 4, 5) from the origin.

13. If the distance of the point (2, 1, k) from the origin is 3, then find k.

14. Find the coordinates of the mid-point of the line joining the points (1, -2, 1) and (-3, 8, 3).

15. Find the coordinates of the centroid of a triangle whose vertices are (2,-3, 1), (1, 0, -1) and (3, 6, 0).

Two marks questions:

1. Find the distance between the points P (1,-3,4) and Q (-4,1,2).

 2. Find the equation of the set of points P such that its distances from the points A (3, 4,-5) and B (-2, 1, 4) are equal.

 3. Find the equation of the set of points P such that PA² + PB² = 2K², where A and B are the points (3, 4, 5) and (-1, 3, -7) respectively.

Three marks questions:

1.Show that the points P (-2, 3, 5), Q (1, 2, 3) and R (7, 0, -1) are collinear.

2. Are the points A(3,6,9), B(10,20,30) and C(25, –41,5),the vertices of a right angled triangle.

3. Show that the points (-1,2,1),(1,-2,5),(4,-7,8) and (2,-3,4) are the vertices of a parallelogram.

4. Show that the points (0,7,-10),(1,6,-6) and (4,9,-6) are the vertices of an isosceles triangle.

5. Find the equation of the set of points P, the sum of whose distances from A (4, 0, 0) and B (-4, 0, 0) is equal to 10.

6. The centroid of a triangle ABC is (1, 1, 1).If the coordinates of A and B are (3,-5, 7) and (-1, 7,-6) respectively, find the coordinates of the point C.

7. Find the coordinates of the point which trisect the line segment joining the points P(4,2,-6) and  Q(10,-16,6).

Five marks questions:

1. Derive the formula to find the coordinates of the point which divides the line segment joining

 thepoints (x1, y1, z1) and (x2,y2,z2) in the ratio m:n internally.

2. Find the coordinates of the point which divides the line segment joining the point (1,-2, 3) and (3,4,-5) in the ratio 2:3 (i) internally and (ii) externally.

3. Using section formula, prove that the three points (-4,6,10),(2,4,6)and (4,0,-2)are collinear.

4. Given that P(3,2,-4),Q(5,4,-6)and R(9,8,-10) are collinear. Find the ratio in which Q divides PR. Also find the coordinates of Q.

5. Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, -8) is divided by the YZ - plane.

11.Find the distances of the point P(-4, 3, 5) from the coordinate axes. 

12.Find the distance formula that the points P(-2, 4, 1), Q(1,2,-5) . 

13.Find the distance between the following pairs of points: P(1,-1, 0) and Q(2, 1, 2) 

14.Find the distance between the points P and Q having coordinates (-2, 3, 1) and (2, 1, 2).

15.Name the octants in which the following points lie. (i) (1, 2, 3) (ii) (-2, 4, 6) (iii) (3,-5, 7) (iv) (3, 4,-7) 

16.Find the image of the following points A, B, C in XY-plane; D, E, F in YZ-plane and P, Q, R in ZX-plane (i) A (1, 0, 2) (ii) B (-3, 0, 0) (iii) E (0,-3, 6) (iv) F (2, 5, 6) (v) R (-5, 4, 3) 

17.Find the distance between the points P (1,-3, 4) and Q (-4, 1, 2). 

18.Name the octants in which the following points lie (i)  (1, 2, 5) (ii)  (-3,-1, 2) (iii) (-3,-1, 2) 

19.Where are the following points: (i) (0, 0,-4) (ii) (0, 3,-2) 

20.Fill in the blanks: 

(i) Z-coordinates of a point lying in XY-plane is ......... 

(ii) The x-axis and y-axis taken together determine a plane known as ....... 

(iii) The coordinates of point on x-axis of the form ................... 

21.Find the image of the point in the specified plane (i) (0, 0,-4) in xy-plane (ii) (-3, 4, 7) in the xy-plane 

22.Find the distance of point (-1,-3, 4) from x, y and z axes. 

23.Find the distance between the points (4, 3,-6) and (-2, 1,-3). 

24.If a point lie on the z-axis, then find its coordinate and y-coordinate. 

25.If a point lies on yz-plane then what is its x-coordinate? 

26.In which plane does the point (4,-3, 0) lie? 

27.A point is on the x-axis. Write its y-coordinate and z-coordinate. 

28.In three dimensional geometry, write the equation of x-axis. 

29.A point is in the XZ-plane. What can you say about its y-coordinate? 

30.Find the coordinates of a point on Y-axis which is at a distance of from the point P(3,-2, 5). 

31.Using distance formula prove that the following points are collinear. A(4,-3,-1), B(5,-7, 6) and C(3, 1,-8) 

32.Find the points on z-axis which are at a distance from the point (1, 2, 3). 

33.Prove that the triangle formed by joining the three points whose coordinates are (1, 2, 3), (2, 3, 1) and (3, 1, 2) is an equilateral triangle. 

34.Find the coordinates of the point which is equidistant from the four points O(0, 0, 0), A(2, 0, 0), B(0, 3, 0) and C(0, 0, 8). 

35.If A(-2, 2, 3) and B(13,-3, 13) are two points. 

36.Are the points A(3, 6, 9), B(10, 20, 30) and C(25,-41, 5), the vertices of a right-angled triangle. 

37.Verify the following (0, 7,-10), (1, 6,-6) and (4, 9,-6) are vertices of an isosceles triangle. 

38.Find the locus of the points which are equidistant from the points (1, 2, 3) and (3, 2,-1).