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<p>The diagram above shows part of the structure of a rock-climbing wall, with a horizontal base $OABC$, a vertical rectangular wall $OADE$ and an inclined rectangular wall $DEFG$. With respect to the origin $O$, the vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ each of length $1$ m, are parallel to $OA$, $OC$ and $OE$ respectively. It is given that $OE=10$m.</p>
<p>The plane $OADE$ has equation $\mathbf{r}\cdot \left( \begin{matrix}
0 \\
1 \\
0 \\
\end{matrix} \right)=p$, and the plane $DEFG$ has equation $\mathbf{r}=\left( \begin{matrix}
0 \\
0 \\
10 \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
3 \\
-2 \\
4 \\
\end{matrix} \right)+\mu \left( \begin{matrix}
0 \\
-1 \\
q \\
\end{matrix} \right)$, where $p$ and $q$ are constants, $\lambda $ and $\mu $ are parameters.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Give a reason why $p=0$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Write down a vector equation of the line passing through the points $D$ and $E$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Show that $q=2$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence, find the obtuse angle between the walls $OADE$ and $DEFG$.<span class="inline-marks">[3]</span></div></li></ol>
<p>A rock climber is at the point $H\left( 2,\,-1,\,12 \right)$.</p>
<ol class="abc-list" start="5"><li><div class="list-item-paragraph">Show that the climber is on the plane $DEFG$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The line $l$ passing through the points $F$ and $G$ has equation $\mathbf{r}=\left( \begin{matrix}
3 \\
-2 \\
14 \\
\end{matrix} \right)+\gamma \left( \begin{matrix}
1 \\
0 \\
0 \\
\end{matrix} \right)$, where $\gamma $ is a parameter.</p>
<ol class="abc-list" start="6"><li><div class="list-item-paragraph">Find the exact shortest distance from the climber to $l$.<span class="inline-marks">[3]</span></div></li></ol>

The diagram above shows part of the structure of a rock-climbing wall, with a horizontal base $OABC$, a vertical rectangular wall $OADE$ and an inclined rectangular wall $DEFG$. With respect to the origin $O$, the vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ each of length $1$ m, are parallel to $OA$, $OC$ and $OE$ respectively. It is given that $OE=10$m.

The plane $OADE$ has equation $\mathbf{r}\cdot \left( \begin{matrix}
0 \\
1 \\
0 \\
\end{matrix} \right)=p$, and the plane $DEFG$ has equation $\mathbf{r}=\left( \begin{matrix}
0 \\
0 \\
10 \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
3 \\
-2 \\
4 \\
\end{matrix} \right)+\mu \left( \begin{matrix}
0 \\
-1 \\
q \\
\end{matrix} \right)$, where $p$ and $q$ are constants, $\lambda $ and $\mu $ are parameters.

Write down a vector equation of the line passing through the points $D$ and $E$.

Hence, find the obtuse angle between the walls $OADE$ and $DEFG$.

A rock climber is at the point $H\left( 2,\,-1,\,12 \right)$.

Show that the climber is on the plane $DEFG$.

The line $l$ passing through the points $F$ and $G$ has equation $\mathbf{r}=\left( \begin{matrix}
3 \\
-2 \\
14 \\
\end{matrix} \right)+\gamma \left( \begin{matrix}
1 \\
0 \\
0 \\
\end{matrix} \right)$, where $\gamma $ is a parameter.

Find the exact shortest distance from the climber to $l$.

Vectors - General

Vectors

H2 Math

International Baccalaureate Higher Level Mathematics

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