Chapter 9.4, Problem 51E

a.

To determine

Verify that vector equation $\left(r-a\right)\cdot \left(r-b\right)=0$ represent a sphere.

Answer to Problem 51E

  ${\left(x-0\right)}^2+{\left(y-0\right)}^2+{\left(z-1\right)}^2=9$

Explanation of Solution

Given information:

The vector equation of a sphere having points, $a=\langle 2,2,2\rangle ,b=\langle -2,-2,0\rangle andr=\langle x,y,z\rangle$ .

Calculation:

Consider the given points,

  $a=\langle 2,2,2\rangle ,b=\langle -2,-2,0\rangle andr=\langle x,y,z\rangle$

Now compute $\left(r-a\right)\cdot \left(r-b\right)=0$ ,

  $\begin{array}{l}\left(r-a\right)\cdot \left(r-b\right)=\langle x-2,y-2,z-2\rangle \cdot \langle x+2,y+2,z\rangle \\ =\left(x-2\right)\left(x+2\right)+\left(y-2\right)\left(y+2\right)+z\left(z-2\right)\\ x^2+y^2+z^2-2z-4-4=0\\ x^2+y^2+z^2-2z+1=9\\ {\left(x-0\right)}^2+{\left(y-0\right)}^2+{\left(z-1\right)}^2=9\end{array}$

Hence, the equation is, ${\left(x-0\right)}^2+{\left(y-0\right)}^2+{\left(z-1\right)}^2=9$ .

b.

To determine

Find the centre and radius of the sphere.

Answer to Problem 51E

  $C\left(0,0,1\right),r=3$

Explanation of Solution

Given information:

The vector equation of a sphere having points, $a=\langle 2,2,2\rangle ,b=\langle -2,-2,0\rangle andr=\langle x,y,z\rangle$ .

Calculation:

Consider the equation,

  ${\left(x-0\right)}^2+{\left(y-0\right)}^2+{\left(z-1\right)}^2=9$

Compare it with standard equation of sphere,

  $\begin{array}{l}{\left(x-a\right)}^2+{\left(y-b\right)}^2+{\left(z-c\right)}^2=r^2\\ C\left(0,0,1\right),r=3\end{array}$

Hence, the centre and radius of the sphere is $C\left(0,0,1\right),r=3$

c.

To determine

Interpret the result of part $\left(a\right)$ geometrically.

Answer to Problem 51E

  

Explanation of Solution

Given information:

The vector equation of a sphere having points, $a=\langle 2,2,2\rangle ,b=\langle -2,-2,0\rangle andr=\langle x,y,z\rangle$ .

Using the fact that the dot product of two vectors is $0$ only if the vectors are perpendicular

Calculation:

Graph the function,

  

d.

To determine

Find the centre and radius of sphere.

Answer to Problem 51E

  $C\left(1,0,7/2\right),r=1.5$

Explanation of Solution

Given information:

The vector equation of a sphere having points, $a=\langle 0,1,3\rangle ,b=\langle 2,-1,4\rangle andr=\langle x,y,z\rangle$ .solve the vector equation for sphere.

Calculation:

Consider the given points,

Now compute $\left(r-a\right)\cdot \left(r-b\right)=0$ ,

  $\begin{array}{l}\left(r-a\right)\cdot \left(r-b\right)=\langle x,y-1,z-3\rangle \cdot \langle x-2,y+1,z-4\rangle \\ =x\left(x-2\right)+y^2-1+\left(z-3\right)\left(z-4\right)\\ x^2+y^2+z^2-2x-7z-1+12\\ x^2+y^2+z^2-2x-7z+1+12.25=-11+1+12.25\\ {\left(x-1\right)}^2+y^2+{\left(z-7/2\right)}^2=2.25\end{array}$

Compare it with standard eqation,

  $\begin{array}{l}{\left(x-a\right)}^2+{\left(y-b\right)}^2+{\left(z-c\right)}^2=r^2\\ C\left(1,0,7/2\right),r=1.5\end{array}$

Hence, the centre and radius sphere is, $C\left(1,0,7/2\right),r=1.5$