Math 22 The Three-Dimensional Coordinate System

The Three-Dimensional Coordinate System

The Distance and Midpoint Formulas

 The distance $d$ between the points $(x_{1},x_{2},x_{3})$ and $(x_{2},y_{2},z_{2})$ is
 
 $d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}}$

Exercises 1 Find the distance between two points

1) $(4,2,3)$ and $(1,2,0)$

Solution:
$d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}}={\sqrt {(1-4)^{2}+(2-2)^{2}+(0-3)^{2}}}={\sqrt {18}}$

2) $(1,2,4)$ and $(2,5,1)$

Solution:
$d={\sqrt {(2-1)^{2}+(5-2)^{2}+(1-4)^{2}}}={\sqrt {19}}$

Midpoint Formula in Space

 The midpoint of the line segment joining the points $(x_{1},x_{2},x_{3})$ and $(x_{2},y_{2},z_{2})$ is
 
 ${\text{Midpoint}}=({\frac {x_{1}+x_{2}}{2}},{\frac {y_{1}+y_{2}}{2}},{\frac {z_{1}+z_{2}}{2}})$

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