Chapter 9.3, Problem 2E

The distance between the point P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is given by the formula d(P, Q) =_______________. The distance between the point P in the figure and the origin is __________. The equation of the sphere centered at P with radius 3 is _______________.

To determine

To fill: The distance between the points $P\left(x_1,y_1,z_1\right)$ and $Q\left(x_2,y_2,z_2\right)$ is given by the formula $d\left(P,Q\right)$ =______. The distance between the point $P$ in the figure and the origin is _____. The equation of the sphere centred at $P$ with radius 3 is _____.

Answer to Problem 2E

The distance between the points $P\left(x_1,y_1,z_1\right)$ and $Q\left(x_2,y_2,z_2\right)$ is given by the formula $d\left(P,Q\right)=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2+{\left(z_2-z_1\right)}^2}$. The distance between the point $P$ in the figure and the origin is $\sqrt{x_1^2+y_1^2+z_1^2}$. The equation of the sphere centred at $P$ with radius 3 is ${\left(x-x_1\right)}^2+{\left(y-y_1\right)}^2+{\left(z-z_1\right)}^2=9$.

Explanation of Solution

The point P with the coordinate $\left(x_1,y_1,z_1\right)$ and point Q with coordinate $\left(x_2,y_2,z_2\right)$.

The coordinates of origin are $\left(0,0,0\right)$ and the coordinates of point P are $\left(x_1,y_1,z_1\right)$.

The distance between two points $A\left(x_a,y_a,z_a\right)$ and $B\left(x_b,y_b,z_b\right)$ is given by the formula,

$D\left(A,B\right)=\sqrt{{\left(x_b-x_a\right)}^2+{\left(y_b-y_a\right)}^2+{\left(z_b-z_a\right)}^2}$

Substitute $x_1$ for $x_a$, $x_2$ for $x_b$, $y_1$ for $y_a$, $y_2$ for $y_b$, $z_1$ for $z_a$ and $z_2$ for $z_b$ in the above formula.

$d\left(P,Q\right)=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2+{\left(z_2-z_1\right)}^2}$

Hence, the distance between the points P and Q is $\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2+{\left(z_2-z_1\right)}^2}$.

Obtain the distance between origin and point P.

The distance between two points $A\left(x_a,y_a,z_a\right)$ and $B\left(x_b,y_b,z_b\right)$  is given by the formula,

$D\left(A,B\right)=\sqrt{{\left(x_b-x_a\right)}^2+{\left(y_b-y_a\right)}^2+{\left(z_b-z_a\right)}^2}$

Substitute $x_1$ for $x_a$, $0$ for $x_b$, $y_1$ for $y_a$, $0$ for $y_b$, $z_1$ for $z_a$ and $0$ for $z_b$ in the above formula in the above formula,

$\begin{array}{c}d\left(O,P\right)=\sqrt{{\left(0-x_1\right)}^2+{\left(0-y_1\right)}^2+{\left(0-z_1\right)}^2}\\ =\sqrt{x_1^2+y_1^2+z_1^2}\end{array}$

Hence, the distance between origin and point P is $\sqrt{x_1^2+y_1^2+z_1^2}$.

Obtain equation of sphere with centre at point P and radius 3.

The sphere with centre $\left(x_c,y_c,z_c\right)$ and radius r is locus of all points $\left(x,y,z\right)$ such that,

${\left(x-x_c\right)}^2+{\left(y-y_c\right)}^2+{\left(z-z_c\right)}^2=r^2$

The coordinates of point P are $\left(x_1,y_1,z_1\right)$.

Substitute 3 for r in the above equation,

$\begin{array}{l}{\left(x-x_1\right)}^2+{\left(y-y_2\right)}^2+{\left(z-z_3\right)}^2={\left(3\right)}^2\\ {\left(x-x_1\right)}^2+{\left(y-y_2\right)}^2+{\left(z-z_3\right)}^2=9\end{array}$

Hence, the equation of sphere with centre $\left(x_1,y_1,z_1\right)$ and radius 3 is ${\left(x-x_1\right)}^2+{\left(y-y_2\right)}^2+{\left(z-z_3\right)}^2=9$.