2022TrialShort answer

Q15 (a) (iii)

2 marks

Question

Show that the line
$\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}15\\-1\\6\end{pmatrix}+\delta\begin{pmatrix}4\\-1\\-1\end{pmatrix}$
is also tangent to the sphere
$(x-5)^2+(y+2)^2+(z-3)^2=38$
by showing that it is perpendicular to the radius of the sphere at $\begin{pmatrix}7\\1\\8\end{pmatrix}$ .

Try the question before revealing the answer.

Solution

(iii) Radius of sphere at $\begin{pmatrix}7\\1\\8\end{pmatrix}$

=\begin{pmatrix}7\\1\\8\end{pmatrix}-\begin{pmatrix}5\\-2\\3\end{pmatrix}

=\begin{pmatrix}2\\3\\5\end{pmatrix}.

Showing

\begin{pmatrix}15\\-1\\6\end{pmatrix}+\delta\begin{pmatrix}4\\-1\\-1\end{pmatrix}

$\perp$ radius.

\begin{pmatrix}4\\-1\\-1\end{pmatrix}\cdot\begin{pmatrix}2\\3\\5\end{pmatrix} =8-3-5

=0.

As direction vector $\cdot$ radius $=0$ .

Line $\perp$ radius.

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Syllabus alignment

Vectors

Vectors and vector equations of linesUse coordinates and vector equations for curves, lines, line segments and spheres in two and three dimensions.
Further operations with three-dimensional vectorsUse vector magnitude, unit vectors and scalar products to solve problems and prove geometric results.

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