A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>The planes $\pi_1$ and $\pi_2$, which meet in the line $l_1$, have equations</p>
<p style="text-align: center">\[\pi_1 : \mathbf{r} \cdot \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix} = 25 \quad \text{and} \quad \pi_2 : x + ky - 2z = -15,\]</p>
<p>where&nbsp;$k$ is a constant.</p>
<p>Another line $l_2$ has equation $\mathbf{r} = \begin{pmatrix} -15 \\ 0 \\ 0 \end{pmatrix} + \beta \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix}, \beta \in \mathbb{R}$.</p>
<ol class="i-list"><li data-list-item-id="imported-item-block-4-item-0" data-list-item-path="imported-item-block-4-item-0"><div class="list-item-paragraph">Determine the position vector of the point on $l_1$ such that the coordinates of this point is independent of $k$. Hence find a vector equation of $l_1$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="imported-item-block-4-item-1" data-list-item-path="imported-item-block-4-item-1"><div class="list-item-paragraph">Determine the possible value(s) of $k$ such that&nbsp;$l_1$ and $l_2$ are skew.<span class="inline-marks">[3]</span></div></li></ol>
<p>Assume that&nbsp;$k = 4$ for the rest of this question.</p>
<ol class="i-list" start="3"><li data-list-item-id="imported-item-block-6-item-0" data-list-item-path="imported-item-block-6-item-0"><div class="list-item-paragraph">Points $A$ and $B$ are on $l_1$ and $l_2$ respectively such that&nbsp;$\overrightarrow{AB}$ is perpendicular to both lines. Show that&nbsp;$\left\vert{} \overrightarrow{AB} \right\vert{} = \sqrt{\frac{p}{2}}$, where&nbsp;$p$ is an integer to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-6-item-1" data-list-item-path="imported-item-block-6-item-1"><div class="list-item-paragraph">Find exactly the sine of the acute angle between $l_2$ and $\pi_1$.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is given further that&nbsp;$l_2$ lies on a third plane that is perpendicular to $l_1$, and $l_2$ intersects $\pi_1$ at point $P$.</p>
<ol class="i-list" start="5"><li data-list-item-id="imported-item-block-8-item-0" data-list-item-path="imported-item-block-8-item-0"><div class="list-item-paragraph">Deduce the shortest distance from $P$ to $l_1$.<span class="inline-marks">[2]</span></div></li></ol>

The planes $\pi_1$ and $\pi_2$, which meet in the line $l_1$, have equations

\[\pi_1 : \mathbf{r} \cdot \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix} = 25 \quad \text{and} \quad \pi_2 : x + ky - 2z = -15,\]

Another line $l_2$ has equation $\mathbf{r} = \begin{pmatrix} -15 \\ 0 \\ 0 \end{pmatrix} + \beta \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix}, \beta \in \mathbb{R}$.

Determine the position vector of the point on $l_1$ such that the coordinates of this point is independent of $k$. Hence find a vector equation of $l_1$.

Determine the possible value(s) of $k$ such that $l_1$ and $l_2$ are skew.

Assume that $k = 4$ for the rest of this question.

Points $A$ and $B$ are on $l_1$ and $l_2$ respectively such that $\overrightarrow{AB}$ is perpendicular to both lines. Show that $\left\vert{} \overrightarrow{AB} \right\vert{} = \sqrt{\frac{p}{2}}$, where $p$ is an integer to be determined.

Find exactly the sine of the acute angle between $l_2$ and $\pi_1$.

It is given further that $l_2$ lies on a third plane that is perpendicular to $l_1$, and $l_2$ intersects $\pi_1$ at point $P$.

Deduce the shortest distance from $P$ to $l_1$.

Finding Points on a Plane

Angle Between a Line and a Plane

Common Perpendicular and Distance Between Skew Lines

Relationships Between Two Lines

Perpendicular Vectors Using the Scalar Product

Line of Intersection of Two Planes

Distance from a Point to a Line

2024 NJC P1 Q12

Vectors

<p>Given two lines ${{l}_{1}}:\mathrm{ }\frac{x}{2}=y=z$ and ${{l}_{2}}:\mathrm{ }\frac{x-2}{2}=\frac{-1-y}{3}$, $z=1$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Explain why lines ${{l}_{1}}$ and ${{l}_{2}}$ are skew.</div></li><li><div class="list-item-paragraph">Find the shortest possible distance between the two lines.</div></li><li><div class="list-item-paragraph">Determine the Cartesian equation of the common perpendicular that intersects with ${{l}_{1}}$ as well as with ${{l}_{2}}$.</div></li></ol>

Given two lines ${{l}_{1}}:\mathrm{ }\frac{x}{2}=y=z$ and ${{l}_{2}}:\mathrm{ }\frac{x-2}{2}=\frac{-1-y}{3}$, $z=1$.

Equations of Lines

2023 CJC Challenge Yourself Q2

<p>The lines ${{l}_{1}}$ and ${{l}_{2}}$ have equations</p>
<p style="text-align: center">${{\mathbf{r}}_{1}}=\left( \begin{matrix}
3 \\
2 \\
0 \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
2 \\
-3 \\
1 \\
\end{matrix} \right)$ and ${{\mathbf{r}}_{2}}=\left( \begin{matrix}
-4 \\
1 \\
-3 \\
\end{matrix} \right)+\mu \left( \begin{matrix}
1 \\
2 \\
4 \\
\end{matrix} \right)$</p>
<p>respectively, where&nbsp;$\lambda $ and $\mu $ are parameters.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find a cartesian equation of the plane containing ${{l}_{1}}$ and the point $\left( 1,-3,-1 \right)$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Show that&nbsp;${{l}_{1}}$ is perpendicular to ${{l}_{2}}$.<span class="inline-marks">[2]</span></div></li><li><ol class="i-list"><li><div class="list-item-paragraph">Find values of $\lambda $ and $\mu $ such that&nbsp;${{\mathbf{r}}_{1}}-{{\mathbf{r}}_{2}}$ is perpendicular to both&nbsp;${{l}_{1}}$ and ${{l}_{2}}$. State the position vectors of the points where the common perpendicular meets ${{l}_{1}}$ and ${{l}_{2}}$.<span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">Find the length of this common perpendicular.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

The lines ${{l}_{1}}$ and ${{l}_{2}}$ have equations

${{\mathbf{r}}_{1}}=\left( \begin{matrix}
3 \\
2 \\
0 \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
2 \\
-3 \\
1 \\
\end{matrix} \right)$ and ${{\mathbf{r}}_{2}}=\left( \begin{matrix}
-4 \\
1 \\
-3 \\
\end{matrix} \right)+\mu \left( \begin{matrix}
1 \\
2 \\
4 \\
\end{matrix} \right)$

respectively, where $\lambda $ and $\mu $ are parameters.

Magnitude of a Vector

Equations of Planes

N2021 P1 Q8

<p>A pair of skew lines ${{l}_{1}}$ and ${{l}_{2}}$ have equations</p>
<p style="text-align: center">$\mathbf{r}=\left( \begin{matrix}
-1 \\
1 \\
3 \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
-2 \\
2 \\
1 \\
\end{matrix} \right)$, $\lambda \in \mathbb{R}$ and $\mathbf{r}=\left( \begin{matrix}
1 \\
3 \\
2 \\
\end{matrix} \right)+\mu \left( \begin{matrix}
3 \\
1 \\
6 \\
\end{matrix} \right)$, $\mu \in \mathbb{R}$</p>
<p>respectively. Suppose the points $P$ and $Q$ lie on ${{l}_{1}}$ and ${{l}_{2}}$ respectively such that $PQ$ is the shortest distance between ${{l}_{1}}$ and ${{l}_{2}}$ , find the length of $PQ$.</p><p style="text-align: right;">[4]</p>
<p>(Hint: What relationship(s) does $\overrightarrow{PQ}$ have with ${{l}_{1}}$ and ${{l}_{2}}$?)</p>

A pair of skew lines ${{l}_{1}}$ and ${{l}_{2}}$ have equations

$\mathbf{r}=\left( \begin{matrix}
-1 \\
1 \\
3 \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
-2 \\
2 \\
1 \\
\end{matrix} \right)$, $\lambda \in \mathbb{R}$ and $\mathbf{r}=\left( \begin{matrix}
1 \\
3 \\
2 \\
\end{matrix} \right)+\mu \left( \begin{matrix}
3 \\
1 \\
6 \\
\end{matrix} \right)$, $\mu \in \mathbb{R}$

respectively. Suppose the points $P$ and $Q$ lie on ${{l}_{1}}$ and ${{l}_{2}}$ respectively such that $PQ$ is the shortest distance between ${{l}_{1}}$ and ${{l}_{2}}$ , find the length of $PQ$.

(Hint: What relationship(s) does $\overrightarrow{PQ}$ have with ${{l}_{1}}$ and ${{l}_{2}}$?)

Shortest Distance Between 2 Skew Lines

<p>The equations of the plane $\pi$, and the lines ${{l}_{1}}$ and ${{l}_{2}}$ are given by</p>
<p style="text-align: center">$\begin{aligned} & \pi :tx-2y+z=13 \\ & {{l}_{1}}:\mathbf{r}=\left( 2\mathbf{i}+\mathbf{j}+3\mathbf{k} \right)+\lambda \left[ 2t\mathrm{ }\mathbf{i}+\left( {{t}^{2}}-1 \right)\mathbf{j}-2\mathbf{k} \right] \\ & {{l}_{2}}:x-4=y,z=5, \\ \end{aligned}$</p>
<p>where&nbsp;$t$ and $\lambda $ are real constants.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Given that the shortest distance from the point P with coordinates $\left( 3,3,-5 \right)$ to $\pi$ is $6$, find the possible values of $t$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Show that&nbsp;${{l}_{1}}$ is parallel to $\pi$, and find a condition of $t$ such that&nbsp;${{l}_{1}}$ is not on $\pi$. </div><div class="list-item-paragraph">For the case where&nbsp;$t=2$,<span class="inline-marks">[3]</span></div><ol class="i-list"><li><div class="list-item-paragraph">show that&nbsp;${{l}_{2}}$ lies on $\pi$,<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">given that&nbsp;$A$ is a point on ${{l}_{1}}$ and $B$ is a point on ${{l}_{2}}$, find the position vectors of $A$ and $B$ such that&nbsp;$AB$ is perpendicular to both&nbsp;${{l}_{1}}$ and ${{l}_{2}}$,<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">find the vector equation of the line of reflection of ${{l}_{1}}$ in $\pi$.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

The equations of the plane $\pi$, and the lines ${{l}_{1}}$ and ${{l}_{2}}$ are given by

$\begin{aligned} & \pi :tx-2y+z=13 \\ & {{l}_{1}}:\mathbf{r}=\left( 2\mathbf{i}+\mathbf{j}+3\mathbf{k} \right)+\lambda \left[ 2t\mathrm{ }\mathbf{i}+\left( {{t}^{2}}-1 \right)\mathbf{j}-2\mathbf{k} \right] \\ & {{l}_{2}}:x-4=y,z=5, \\ \end{aligned}$

where $t$ and $\lambda $ are real constants.

Math Question Bank