(– , , ). . Show that the points (– , , ), ( , , ) and ( , , – ) are collinear. .
Verify the following: ( , , – ), ( , , – ) and ( , , – ) are the vertices of an isosceles triangle. ( , , ), (– , , ) and (– , , ) are the vertices of a right angled triangle. (iii) (– , , ), ( , – , ), ( , – , ) and ( , – , ) are the vertices of a parallelogram. .
Find the equation of the set of points which are equidistant from the points ( , , ) and ( , , – ). . Find the equation of the set of points P, the sum of whose distances from A ( , , ) and B (– , , ) is equal to . .
Section Formula In two dimensional geometry, we have learnt how to find the coordinates of a point dividing a line segment in a given ratio internally. Now, we extend this to three dimensional geometry as follows: Let the two given points be P( x , y , z ) and Q ( x , y , z ). Let the point R ( x , y , z ) divide PQ in the given ratio m : n internally. Draw PL, QM and RN perpendicular to MATHEMATICS the XY-plane.
Obviously PL || RN || QM and feet of these perpendiculars lie in a XY-plane. The points L, M and N will lie on a line which is the intersection of the plane containing PL, RN and QM with the XY-plane. Through the point R draw a line ST parallel to the line LM. Line ST will intersect the line LP externally at the point S and the line MQ at T, as shown in Fig .
. Also note that quadrilaterals LNRS and NMTR are parallelograms. The triangles PSR and QTR are similar. Therefore, PR SP SL PL NR PL QR QT QM TM QM NR m z z z z – – – – – – This implies