46. Derive an expression for the coordinates of a point which divides the line joining the
points A(x1,y1,z1,) and B(x2, Y2,Z2) internally in the ratio m:n.
Answers
Answer:
Let the two given points be P (x1, y1, z1,) and Q (x2, y2, z2). Let the point R (x, y, z) divide PQ in the given ratio m : n internally, Draw PL, QM and RN perpendicular to 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, m/n = PR/QR = SP/QT = (SL - PL)/(QM - TM) = (NR - PL)/(QM - NR) = (Z - z1)/(Z2 - z) This implies z = (mz2 + nz1)/(m + n) Similarly, by drawing perpendiculars to the XZ and YZ-planes, we get y = (my2 + ny1)/(m + n) and x = (mx2 + nx1)/(m + n) Hence, the coordinates of the point R which divides the line segment joining two points P(x1, y1, z1) and Q(x2, y2, z2) internally in the ratio m: n are R(x, y, z) = ((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n), (mz2 + nz1)/(m + n)).Read more on Sarthaks.com - https://www.sarthaks.com/595988/derive-an-expression-for-the-coordinates-of-point-that-divides-the-line-joining-the-points
Answer:
Derive an expression for the coordinates of a point that divides the line joining the points A(x1, y1, z1,) and B (x2, y2, z2.) internally in the ratio m : n. Hence, find the coordinates of the midpoint of AB where A = (1, 2, 3) and B = (5, 6, 7)