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<p>${{\Pi }_{1}}:\mathbf{r}\cdot \left( \begin{matrix}
   -2  \\
   -3  \\
   1  \\
\end{matrix} \right)=7$, ${{\Pi }_{2}}:2x+3y-z=4$<br></p>

${{\Pi }_{1}}:\mathbf{r}\cdot \left( \begin{matrix}
   -2  \\
   -3  \\
   1  \\
\end{matrix} \right)=7$, ${{\Pi }_{2}}:2x+3y-z=4$


Distance from a Point to a Plane

Angle Between Two Planes

Relationships Between Two Planes

Vectors

<p>${{\Pi }_{1}}:\mathbf{r}=\left( \begin{matrix}1  \\3  \\1  \\\end{matrix} \right)+\lambda \left( \begin{matrix} 2  \\-2  \\ 1  \\\end{matrix} \right)+\mu \left( \begin{matrix}1  \\ 5  \\3  \\\end{matrix} \right);\lambda ,\mu \in \mathbb{R}$, </p>
<p>${{\Pi }_{2}}:(2t+3s)\mathbf{i}+(1-2t+3s)\mathbf{j}+(-2+t+4s)\mathbf{k};s,t\in \mathbb{R}$</p>

${{\Pi }_{1}}:\mathbf{r}=\left( \begin{matrix}1  \\3  \\1  \\\end{matrix} \right)+\lambda \left( \begin{matrix} 2  \\-2  \\ 1  \\\end{matrix} \right)+\mu \left( \begin{matrix}1  \\ 5  \\3  \\\end{matrix} \right);\lambda ,\mu \in \mathbb{R}$, 

${{\Pi }_{2}}:(2t+3s)\mathbf{i}+(1-2t+3s)\mathbf{j}+(-2+t+4s)\mathbf{k};s,t\in \mathbb{R}$

Equations of Planes

<p>Find the relationship and the acute angles between the two planes. If the planes are parallel to each other, find the perpendicular distance between the planes, otherwise, find the equation of line which two planes intersect.</p>
<p>${{\Pi }_{1}}:\mathbf{r}=\left( \begin{matrix} 1\\ 0\\ -3 \\\end{matrix} \right)+\lambda \left( \begin{matrix} -2 \\1\\ 0 \\\end{matrix} \right)+\mu \left( \begin{matrix} 1\\ -2\\ 3 \\\end{matrix} \right);\lambda ,\mu \in \mathbb{R}$, </p>
<p>${{\Pi }_{2}}:(-1+2t-3s)\mathbf{i}+(-3-t+s)\mathbf{j}+(5+2t-s)\mathbf{k};s,t\in \mathbb{R}$</p>

Find the relationship and the acute angles between the two planes. If the planes are parallel to each other, find the perpendicular distance between the planes, otherwise, find the equation of line which two planes intersect.

${{\Pi }_{1}}:\mathbf{r}=\left( \begin{matrix} 1\\ 0\\ -3 \\\end{matrix} \right)+\lambda \left( \begin{matrix} -2 \\1\\ 0 \\\end{matrix} \right)+\mu \left( \begin{matrix} 1\\ -2\\ 3 \\\end{matrix} \right);\lambda ,\mu \in \mathbb{R}$, 

${{\Pi }_{2}}:(-1+2t-3s)\mathbf{i}+(-3-t+s)\mathbf{j}+(5+2t-s)\mathbf{k};s,t\in \mathbb{R}$

Line of Intersection of Two Planes

<p>Find the relationship and the acute angles between the two planes. If the planes are parallel to each other, find the perpendicular distance between the planes, otherwise, find the equation of line which two planes intersect.</p>
<p>${{\Pi }_{1}}:x+2y-5z=13$, ${{\Pi }_{2}}:x+3z=-2$</p>

${{\Pi }_{1}}:x+2y-5z=13$, ${{\Pi }_{2}}:x+3z=-2$

<p>${{\Pi }_{1}}:\mathbf{r}\cdot\left(\begin{matrix}3\\-6\\1\\\end{matrix}\right)=1$; ${{\Pi}_{2}}:2x-3y+z=10$</p>

${{\Pi }_{1}}:\mathbf{r}\cdot\left(\begin{matrix}3\\-6\\1\\\end{matrix}\right)=1$; ${{\Pi}_{2}}:2x-3y+z=10$

<p>Find the relationship and the acute angles between the two planes. If the planes are parallel to each other, find the perpendicular distance between the planes, otherwise, find the equation of line which two planes intersect.</p>
<p>${{\Pi }_{1}}:\mathbf{r}\cdot\left(\begin{matrix}1\\-2\\1  \\\end{matrix}\right)=13$, ${{\Pi }_{2}}:-x+2y-z=5$</p>

${{\Pi }_{1}}:\mathbf{r}\cdot\left(\begin{matrix}1\\-2\\1  \\\end{matrix}\right)=13$, ${{\Pi }_{2}}:-x+2y-z=5$

<p>The line ${{l}_{1}}$ passes through the point $A$, whose position vector is $-\mathbf{i}+2\mathbf{j}$, and is parallel to the vector $\mathbf{i}+\mathbf{k}$. The line ${{l}_{2}}$ passes through the point $B$, whose position vector is $\mathbf{i}+\mathbf{j}+3\mathbf{k}$, and is parallel to the vector $\mathbf{j}+\mathbf{k}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the lines ${{l}_{1}}$ and ${{l}_{2}}$ are skew.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the position vector of the point $N$ on ${{l}_{2}}$ such that $AN$ is perpendicular to ${{l}_{2}}$.</div><p style="text-align: right;">[3]</p></li></ol>
<p>The plane $\prod $ contains ${{l}_{2}}$ and is perpendicular to $AN$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find a vector equation for $\prod $ in the form $\mathbf{r}=\mathbf{u}+a\mathbf{v}+\beta \mathbf{w}$, where $\mathbf{v}$ and $\mathbf{w}$ are perpendicular vectors.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">The point $X$ varies in such a way that the mid-point of $AX$ is always in $\prod $. Show that $X$ lies on a plane parallel to $\prod $.</div><p style="text-align: right;">[4]</p></li></ol>

The line ${{l}_{1}}$ passes through the point $A$, whose position vector is $-\mathbf{i}+2\mathbf{j}$, and is parallel to the vector $\mathbf{i}+\mathbf{k}$. The line ${{l}_{2}}$ passes through the point $B$, whose position vector is $\mathbf{i}+\mathbf{j}+3\mathbf{k}$, and is parallel to the vector $\mathbf{j}+\mathbf{k}$.

The plane $\prod $ contains ${{l}_{2}}$ and is perpendicular to $AN$.

Perpendicular Projection and Reflection in a Line

Relationships Between Two Lines

Ratio Theorem and Section Formula

2016 RI P1 Q11 [Modified]

<p>The plane ${{p}_{1}}$ has equation $2x-y+3z=0$ and the line $l$ has equation $x=\frac{y-1}{a}=\frac{z-3}{b}$, where $a$ and $b$ are non-zero constants. The points $A$ and $B$ have position vectors $9\mathbf{i}+2\mathbf{j}+4\mathbf{k}$ and $\mathbf{j}+3\mathbf{k}$ respectively.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Given that line $l$ is parallel to plane ${{p}_{1}}$, express $a$ in terms of $b$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the position vector of the foot of perpendicular from $A$ to plane ${{p}_{1}}$ and hence find the shortest distance from $A$ to plane ${{p}_{1}}$.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">Find the length of projection of $\overrightarrow{OB}$ on the plane ${{p}_{1}}$.</div><p style="text-align: right;">[3]</p></li></ol>
<p>Another plane ${{p}_{2}}$ has equation $cx-y+3z=d$.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Given that $c=3$ and $d=-1$, find a vector equation of the line of intersection between ${{p}_{1}}$ and ${{p}_{2}}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the value of $c$ and the range of values of $d$ such that planes ${{p}_{1}}$ and ${{p}_{2}}$ are parallel and non-intersecting.</div><p style="text-align: right;">[2]</p></li></ol>

The plane ${{p}_{1}}$ has equation $2x-y+3z=0$ and the line $l$ has equation $x=\frac{y-1}{a}=\frac{z-3}{b}$, where $a$ and $b$ are non-zero constants. The points $A$ and $B$ have position vectors $9\mathbf{i}+2\mathbf{j}+4\mathbf{k}$ and $\mathbf{j}+3\mathbf{k}$ respectively.

Another plane ${{p}_{2}}$ has equation $cx-y+3z=d$.

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