(adsbygoogle = window.adsbygoogle || []).push({}); The three points (1,1,-1), (3,3,2) and (3,-1,-2) determine a plane. Find:

A vector normal to the planeThe equation of the planeThe distance of the plane from the origin

For part 1 i just said let:

[tex]

\begin{array}{l}

\overrightarrow {OA} = < 1,1, - 1 > \\

\overrightarrow {OB} = < 3,3,2 > \\

\overrightarrow {OC} = < 3, - 1, - 2 > \\

\end{array}

[/tex]

then:

[tex]

\begin{array}{l}

\overrightarrow {AB} = < 2,2,3 > \\

\overrightarrow {AC} = < 2, - 2, - 1 > \\

\end{array}

[/tex]

Since the vectors AB and AC both lie on the plane, i just need to find a vector normal to both.

[tex]

\overrightarrow {AB} \times \overrightarrow {AC} = < 4,8, - 8 >

[/tex]

For the equation of the plane i just said

[tex]

\left( {\vec{p} - \left( {\begin{array}{*{20}c}

1 \\

1 \\

{ - 1} \\

\end{array}} \right)} \right) \cdot \left( {\begin{array}{*{20}c}

4 \\

8 \\

{ - 8} \\

\end{array}} \right) = 0

[/tex]

Where vector p is some point on the plane.

For the final part, i think i did it correctly, but not sure if it is the most efficient way of doing it. I assumed that the question was supposed to say theperpendiculardistance.

What i did was find the projection of OA (since A lies on the plane) onto a vector which is normal to the plane, n, and then calculate its magnitude.

[tex]

\begin{array}{l}

{\mathop{\rm proj}\nolimits} _{\overrightarrow n } \overrightarrow {OA} = \frac{{\overrightarrow {OA} \cdot \overrightarrow n }}{{\overrightarrow n .\overrightarrow n }}\overrightarrow n = \frac{{\overrightarrow {OA} \cdot \overrightarrow n }}{{\left| {\overrightarrow n } \right|}}\widehat{n} \\

\therefore \left| {{\mathop{\rm proj}\nolimits} _{\overrightarrow n } \overrightarrow {OA} } \right| = \frac{{\overrightarrow {OA} \cdot \overrightarrow n }}{{\left| {\overrightarrow n } \right|}} = 5/3 \\

\end{array}

[/tex]

Does that look ok? Its mainly the last part i was a bit iffy about; i think ive done parts 1 and 2 correctly, unless ive made errors in my calculations.

Thanks in advance,

Dan.

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# Homework Help: Points On A Plane

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