Vectors, EXT2 EQ-Bank 26

Let \(S\) be a sphere with equation

\begin{align*}
\left|r-\left(\begin{array}{c} 2 \\ -3 \\ 3
\end{array}\right)\right|=15
\end{align*}

Show the point \(P(1,-1,2)\) lies outside the sphere \(S\). (2 marks)

--- 6 WORK AREA LINES (style=lined) ---

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\(\text{Centre of circle} \ (C)=(2,-3,3)\)

\(\text{Radius}=15 \ \text{(given)}\)

\(\text{Find distance from} \ C \text { to } P(1,-1,2):\)

\(\operatorname{dist}=\sqrt{(1-2)^2+(-1+3)^2+(2-3)^2}=\sqrt{6}\)

\(\text{Since \(\ \sqrt{6}<15, P\) lies inside sphere.}\)

Show Worked Solution

\(\text{Centre of circle} \ (C)=(2,-3,3)\)

\(\text{Radius}=15 \ \text{(given)}\)

\(\text{Find distance from} \ C \text { to } P(1,-1,2):\)

\(\operatorname{dist}=\sqrt{(1-2)^2+(-1+3)^2+(2-3)^2}=\sqrt{6}\)

\(\text{Since \(\ \sqrt{6}<15, P\) lies inside sphere.}\)