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# In three dimensions, the location of a point can be represented by the ordered triple (x, y, z). a) Find the coordinates of the midpoint of the line segment with endpoints A(2, 3, 1) and B(6, 7, 5). b) Write an expression for the coordinates of the midpoint of the line segment with endpoints $\left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$ and $\left( x _ { 2 } , y _ { 2 } , z _ { 2 } \right)$

Solution

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$\textbf{a)}$

Use the same concept as finding the midpoint between two points in two dimensions by finding the average (or mean) of the $x$-coordinates, $y$-coordinates, and $z$-coordinates:

$\left(\dfrac{2+6}{2},\dfrac{3+7}{2},\dfrac{1+5}{2}\right)= \left(\dfrac{8}{2},\dfrac{10}{2},\dfrac{6}{2}\right)=\color{#c34632}(4,5,3)$

$\textbf{b)}$

Using what we used in part a, the expression is:

$\color{#c34632}M(x,y,z)=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2},\dfrac{z_1+z_2}{2}\right)$

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