Question

Refer to the figure.

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

Short answer

The distance between the point $P\left(x_1,y_1,z_1\right)$ and $Q\left(x_2,y_2,z_2\right)$ is given by the formula $d(P,Q)=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2+{\left(z_2-z_1\right)}^2}$ The distance between the point in the figure $P$and the origin is $\sqrt{38}$ The equation of the sphere centered at $P$ with radius 3 is ${(x-5)}^2+{(y-2)}^2+{(z-3)}^2=9$

Step-by-step solution

Step 1. Given information.

The distance between the point$P\left(x_1,y_1,z_1\right)$and$Q\left(x_2,y_2,z_2\right)$is given by the formula

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

From the given figure, the coordinates of the point $\text{P=(5,2,3)}$.

And, another point is the origin and is given by $O(0,0,0)$.

Step 2. The distance between the point P in the figure and the origin is 

$\begin{array}{l}d(P,O)=\sqrt{{(5-0)}^2+{(2-0)}^2+{(3-0)}^2}\\ d(P,O)=\sqrt{25+4+9}\\ \therefore d(P,O)=\sqrt{38}.\end{array}$

The equation of the sphere centered at $P(5,2,3)$with radius 3 is given by,

$\begin{array}{l}{(x-k)}^2+{(y-h)}^2+{(z-l)}^2=r^2\\ {(x-5)}^2+{(y-2)}^2+{(z-3)}^2=9\end{array}$