1. Find the distance between the points (2, 3, 1) and (-3, 1, -5)

Sol : Given Points:  A  =  ( 2, 3, 1 )

                                    B  =  ( -3,  1, -5 )

2. Show that the points A ( 2, -1,  3 ), B ( 4, -2, 1 ), C ( 4, 5, -7) and D ( 2, 6, -5 ) form a parallelogram

Sol: Given Points

A    =   ( 2, -1, 3 )

B    =   ( 4, -2, 1 )

C    =   ( 4,  5, -7 )

D    =  ( 2,  6,  -5 )

3. Show that the points A(0, 1, 2), B(2, -1, 3) and C(1, -3, 1) form a right angled isosceles triangle

Sol : Given points;  A   =  ( 0,  1,  2 )

                                    B  =  ( 2,  -1,  3)

4. Show that the points A ( 3,  -2,  4), B(1,  1,  1) and C(-1,  4,  -2) are Collinear

sol: Given Points A  =  (3,  -2,  4)

                                B  =  ( 1,  1,  1)

                                C  =  (-1,  4, -2)

5. Show that the four points A (-2,  4,  1) B (-1,  5,  5), C (2,  2,  5) and D  (1,  1,  1) form a Square

Sol : Given points A  =  (-2,  4,  1), B  =  (-1,  5,  5), C  =  (2,  2,  5), D  =  (1,  1,  1)

6. Find the coordinates of the point which divides the join of A(2,  -1,  4) and B(4,  3,  2) in the ratio 2 : 3
Sol : Given points A  =  (2,  -1,  4) B  =  (4,  3,  2)
Given ratio m  :  n  =  2 : 3

7. Find the ratio in which the line joining the points (1,  2,  3) and (-3,  4,  -5) is divided by the
xy - plane.
Sol: Given points (1,  2,  3) and (-3,  4,  -5)

Required Ratio - z₁  :   z₂
   - 3   :  - 5
     3    :    5

8. Find the Centroid of the triangle formed by the points (7,  - 4,  7), (1,  - 6,  10), (5  - 1,  1)
Sol : Given points A  =  (7, - 4,  7) B = (1, - 6,  10) C = (5, - 1,  1)

9.Find the centroid of the tetrahedron formed by the points (2, 3, -4), (-3, 3, -2), (-1, 4, 2), (3, 5, 1)
Sol: Given Points
A  =  (2,  3,  -4)
B  =  (-3,  3,  -2)
C  =  (-1,  4,  2)
D  =  (3,  5,  1)

10. If (2, 4, -1), (3, 6, -1), (4, 5, 1) are the consecutive vertices of a parallelogram respectively, find its fourth vertex.
Sol. Given Three Vertices A = (2, 4, -1)
B = (3, 6, -1)
C = (4, 5, 1)
Let the fourth vertex be : D
  
     D  =  ( 2  +  4  - 3,  4  +  5  -  6,  - 1  +  1  +  1 )
   Fourth Vertex: D  =  ( 3,  3,  1 )

11. If (3,  2,  -1), (4,  1,  1), (6,  2,  5) are three vertices and (4,  2,  2) is the centroid of a tetrahedron, find the fourth vertex of the tetrahedron.
Sol: Given Vertices: (3,  2,  -1), (4,  1,  1), (6,  2,  5)
         Given Centroid: (4,  2,  2)
   Let the fourth vertex be :  ( x₄,  y₄,  z₄ )