A Note The coordinates of the origin O are (0,0,0). The coordinates of any point · Part 3

Example Are the points A ( , , ), B ( , , ) and C ( , – , ), the vertices of a right angled triangle? Solution By the distance formula, we have AB = ( – ) + ( – ) + ( – ) = + + = BC = ( – ) + (– – ) + ( – ) = + + = INTRODUCTION TO THREE DIMENSIONAL GEOMETRY CA = ( – ) + ( + ) + ( – ) = + + = We find that CA + AB ≠ BC . Hence, the triangle ABC is not a right angled triangle. Example Find the equation of set of points P such that PA + PB = k , where A and B are the points ( , , ) and (– , , – ), respectively. Solution Let the coordinates of point P be ( x , y, z). Here PA = ( x – ) + ( y – ) + ( z – ) PB = ( x + ) + ( y – ) + ( z + ) By the given condition PA + PB = k , we have ( x – ) + ( y – ) + ( z – ) + ( x + ) + ( y – ) + ( z + ) = k i.e., x + y + z – x – y + z = k – . EXERCISE . . Find the distance between the following pairs of points: ( , , ) and ( , , ) (– , , ) and ( , , – ) (iii) (– , , – ) and ( , – , ) (iv) ( , – , ) and