Let the line segment joining the points P (-2, 3, 5) and Q (1, -4, 6) be PQ.
(i) 2: 3 internally
By using section formula,
We know that 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 is given by:
\( \dfrac{mx_2+nx_1}{m+n}, \dfrac{my_2+ny_1}{m+n}, \dfrac{mz_2+nz_1}{m+n}\)
We have, the point R dividing the line segment joining the points (-2,3,5) and (1,-4,6) internally in the ratio 2:3
\( x= \dfrac{2(1)+3(-2)}{2+3}\)
\( y= \dfrac{2(-4)+3(3)}{2+3}\)
\( z= \dfrac{2(6)+3(5)}{2+3}\)
On solving we get,
\( x=\dfrac{-4}{5}\)
\( x=\dfrac{1}{5}\)
\( x=\dfrac{27}{5}\)
Therefore, the coordinates we obtain are \((\dfrac{-4}{5}, \dfrac{1}{5}, \dfrac{27}{5})\)
(ii) 2: 3 externally
By using section formula,
We know that the coordinates of the point R which divides the line segment joining two points P (x1, y1, z1) and Q (x2, y2, z2) externally in the ratio m: n is given by:
\( \dfrac{mx_2+nx_1}{m+n}, \dfrac{my_2+ny_1}{m+n}, \dfrac{mz_2+nz_1}{m+n}\)
We have, the point R dividing the line segment joining the points (-2,3,5) and (1,-4,6) externally in the ratio 2:3
\( x= \dfrac{2(1)+3(-2)}{2-3}\)
\( y= \dfrac{2(-4)+3(3)}{2-3}\)
\( z= \dfrac{2(6)+3(5)}{2-3}\)
On solving we get,
x = -8
y = 17
z = 3
Therefore, the coordinates we obtain are, (-8,17,3)
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