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
 
 ${\text{Midpoint}}=({\frac {x_{1}+x_{2}}{2}},{\frac {y_{1}+y_{2}}{2}},{\frac {z_{1}+z_{2}}{2}})$

Exercises 2 Find the midpoint of two points below:

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

Solution:
${\text{Midpoint}}=({\frac {x_{1}+x_{2}}{2}},{\frac {y_{1}+y_{2}}{2}},{\frac {z_{1}+z_{2}}{2}})=({\frac {4+1}{2}},{\frac {2+2}{2}},{\frac {3+0}{2}})=({\frac {5}{2}},2,{\frac {3}{2}})$

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

Solution:
${\text{Midpoint}}=({\frac {x_{1}+x_{2}}{2}},{\frac {y_{1}+y_{2}}{2}},{\frac {z_{1}+z_{2}}{2}})=({\frac {1+2}{2}},{\frac {2+5}{2}},{\frac {4+1}{2}})=({\frac {3}{2}},{\frac {7}{2}},{\frac {5}{2}})$

Standard Equation of a Sphere

 The standard equation of a sphere with center at $(h,k,j)$ and radius $r$ is:
 
 $(x-h)^{2}+(y-k)^{2}+(z-j)^{2}=r^{2}$

Exercises 3 Find the equation of the sphere that has:

1) Center: $(1,2,0)$ and radius: $5$

Solution:
$(x-h)^{2}+(y-k)^{2}+(z-j)^{2}=r^{2}$
So, $(x-1)^{2}+(y-2)^{2}+(z-0)^{2}=5^{2}$
Therefore, $(x-1)^{2}+(y-2)^{2}+z^{2}=25$

2) Center: $(-1,2,4)$ and radius: $2$

Solution:
$(x-h)^{2}+(y-k)^{2}+(z-j)^{2}=r^{2}$
So, $(x-(-1))^{2}+(y-2)^{2}+(z-4)^{2}=2^{2}$
Therefore, $(x+1)^{2}+(y-2)^{2}+(z-4)^{2}=4$

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