Question

The coordinates of the point which divides the line segment joining the points (1, –2, 3) and (3, 4, –5) internally in the ratio 2:3 is (x, y, z). Find the value of x+y+z
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Answer 1

The coordinates of the point P which divides the line segment joining two points $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ internally in the ratio m:n are $(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n},\frac{m{z}_2+n{z}_1}{m+n})$
Let P(x, y, z) be the point which divides line segment joining A(1, –2, 3) and B(3, 4, –5) internally in the ratio 2:3 therefore,
$x=\frac{2\left(3\right)+3\left(1\right)}{2+3}=\frac{9}{5}$
$y=\frac{2\left(4\right)+3(-2)}{2+3}=\frac{2}{5}$
$z=\frac{2(-5)+3\left(3\right)}{2+3}=\frac{-1}{5}$
$\Rightarrow$ x+y+z = $\frac{9}{5}+\frac{2}{5}+\frac{-1}{5}$
= 2