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QUESTION


Consider the points $L=(1,1,1),\ M=(1,-1,2)$ and $N=(-1,2,3)$.

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\item[(i)]
Write down the vectors $\mathbf{u}=LM$ and $\mathbf{v}=LN$ and
find their lengths.

\item[(ii)]
Calculate the dot product $\mathbf{u.v}$ and the angle $\theta$
between $\mathbf{u}$ and $\mathbf{v}$.

\item[(iii)]
Compute the cross product $\mathbf{u}\times\mathbf{v}$ and use it
to write down the vector equation of the plane $\Pi_1$ containing
the three point $L,M,n$. What is the equation of the plane in
terms of $x,y,z$ coordinates?

\item[(iv)]
Write down the vector equation of the plane $\Pi_2$ parallel to
$\Pi_1$ and passing through the origin. Find the distance between
the planes $\Pi_1$ and $\Pi_2$.

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ANSWER


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\item[(i)]
$\mathbf{u}=(0,-2,1),\ |\mathbf{u}|=\sqrt{5}\ \
\mathbf{v}=(-2,1,2),\ |\mathbf{v}|=3$

\item[(ii)]
$\mathbf{u.v}=0$ so $\theta=\frac{\pi}{2}$

\item[(iii)]
$\mathbf{u}\times\mathbf{v}=(-5,-2,-4)$

$\Pi_1$ has equation $(-5,-2,-4).(\mathbf{x}-(1,1,1))=0$ so
$5x+2y+4z=11$

\item[(iv)]
$\Pi_2$ has equation
$\mathbf{v}.\left(\begin{array}{c}5\\2\\4\end{array}\right)=5x+2y+4z=0$

Distance between the two planes =$|r(\mathbf{u}\times\mathbf{v})|$
where $(-5r,-2r,-4r)$ lies on $\Pi_1$ so $r=\frac{-11}{45}$ hence
$|r(\mathbf{u}\times\mathbf{v}|=\frac{11}{\sqrt{45}}$.

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