Ratio in which the xy-plane divides the join of (1, 2, 3) and (4, 2, 1) is

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3: 1 \text{ externally } \text{ Suppose the XY - plane divides the line segment joining the points P } \left( 1, 2, 3 \right) \text{ and Q } \left( 4, 2, 1 \right) \text{ in the ratio k: 1 } . \text{ Using the section formula, the coordinates of the point of intersection are given by } \left( \frac{k\left( 4 \right) + 1}{k + 1}, \frac{k\left( 2 \right) + 2}{k + 1}, \frac{k\left( 1 \right) + 3}{k + 1} \right) \text{ The Z - coordinate of any point on the XY - plane is zero }. \Rightarrow \frac{k\left( 1 \right) + 3}{k + 1} = 0 \Rightarrow k + 3 = 0 \Rightarrow k = - 3 = - $\frac{3}{1}$ \text{ Thus, the XY - plane divides the line segment joining the given points in the ratio 3: 1 externally } .

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