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 (x_{1}, y_{1}, z_{1}) and Q (x_{2}, y_{2}, z_{2}) 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 (x_{1}, y_{1}, z_{1}) and Q (x_{2}, y_{2}, z_{2}) 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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