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Question

Find the coordinates of the point which divides the line segment joining the points (2,3,5) and (1,4,6) in the ratio
(i) 2:3 internally
(ii) 2:3 externally

Solution
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(i) The coordinates of point R that divides the line segment joining points P(x1,y1,z1) and Q(x2,y2,z2) internally in the ratio m:n are
(mx2+nx1m+n,my2+ny1m+n,mz2+nz1m+n)
Let R(x,y,z) be the point that divides the line segment joining points (2,3,5) and (1,4,6) internally in the ratio 2:3
x=2(1)+3(2)2+3,y=2(4)+3(3)2+3 and z=2(6)+3(5)2+3
i.e., x=45,y=15and z=275
Thus the coordinates of the required point are (45,15,275)
(ii) The coordinates of point R that divides the line segment joining points P(x1,y1,z1) and Q(x2,y2,z2) externally in the ratio m:n are
(mx2nx1mn,my2ny1mnmz2nz1mn)
Let R(x,y,z) be the point that divides the line segment joining points (2,3,5) and (1,4,6) externally in the ratio 2:3
x=2(1)3(2)23,y=2(4)3(3)23 and z=2(6)3(5)23
i.e.,x=8,y=17 and z=3
Thus the coordinates of the required point are (8,17,3)

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