THREE DIMENSIONAL GEOMETRY

1. A line passing through 2, 1,3 and is perpendicular to the line 2 2r i j k i j k and 2 3 2 2r i j k i j k .Obtain

its equation.(Ans: 2 3 6 3 6r i j k i j k ) 2. Show that the line 1 2 3

2 3 4x y z and 4 1

5 2x y z intersect. Find

their point of intersection. (Ans: Point of intersection 1, 1, 1 ). 3. Find the foot of the perpendicular from the point 0,2,3 on the

line 3 1 45 2 3

x y z . Also, find the length of the perpendicular. (Ans: 2,3, 1 , length = 21 units)

PLANE

4. A plane meets the coordinate axes in A,B,C such that the centroid of triangle ABC is the point , ,p q r . Show that the equation of the plane is

3x y zp q r .

5. The foot of perpendicular drawn from the origin to the plane is 4, 2, 5 . Find the equation of the plane.(Ans: 4 2 5 45x y z )

6. Find the intercepts made on the coordinate axes by the plane 2 2 3x y z and find also the direction cosines of the normal to the plane.

(Ans: 3 3 2 1 2,3, ; , ,2 2 3 3 3

) 7. Find the equation of the plane which bisects the line joining the points

1,2,3 and 3, 5,6 at right angles. (Ans: 4 7 3 28 0x y z ) 8. Find the equation of the plane passing through the point 1, 2,1 and

perpendicular to the line joining the points 3,1,2 and 2,3,4 . Find also the perpendicular distance of the origin from this plane.

(Ans: 5 2 2 1 33 33 33 33

r i j k ; Perpendicular distance 1

33)

9. Find the equation of the plane through the points 2,1 1 and 1,3, 4 and perpendicular to the plane 2 4 10x y z . (Ans: 18 17 4 49x y z )

10. Find the equation of the plane through the point 1, 4, 2 and parallel to the plane 2 3 7x y z .(Ans: 2 3 8 0x y z )

11. Find the direction ratios of the normal to the plane passing through the point 2,1,3 and the line of intersection of the planes 2 3x y z and 2 5x y z . (Ans:13 6 35 0x y z ).

12. Find the equation of the plane which is perpendicular to the plane 5 3 6 8 0x y z and which contains the line of intersection of the planes

2 3 4 0x y z and 2 5 0x y z .(Ans:51 15 50 173 0x y z ) 13. Find the cartesian as well as vector equations of the planes through the

intersection of the planes 2 6 12 0r i j and 3 4 0r i j k which are at a unit distance from the origin.(Ans: 2 2 3 0r i j k ).

14. If the points 1,1, and 3,0,1 be equidistant from the plane 3 4 12 13 0r i j k , find the value of . (Ans: 73 )

15. Find the distance between parallel planes 4 0x y z and 5 0x y z . (Ans: 1

3)

16. Find the distance of the point 2,3,5 from the xy -plane. 17. Find the equations of the line passing through the point 3,0,1 and parallel to

the planes 2 0x y and3 0y z . (Ans: 3 12 1 3

x y z ) 18. Find the equation of the plane through the points 1,0, 1 , 3,2, 2 and parallel

to the line 1 1 21 2 3

x y z .(Ans: 4 2 6 0x y z ) 19. Find the equation of the plane passing through the intersection of the planes

4 10x y z and 4x y z and parallel to the line with direction ratios proportional to 2, 1, 1. Find also the perpendicular distance of 1,1,1 from this plane. (Ans: 3 2

5).

20. Show that the lines 3r i j k i j and 4 2 3r i k i k are coplanar. Also, find the plane containing these two lines. (Ans: 3 9 2 14r i j k .)

21. Find the equation of the plane passing through the point 0,7, 7 and containing the line 1 3 2

3 2 1x y z . (Ans: 0x y z ).

22. If 4 4 0x y z is the equation of the plane through the origin that contains the line 1 1

2 3 4x y z , find the value of .(Ans: 5 ).