A-Level H2 Math | 5 Essential Questions
Vectors
A vector is a quantity that has both magnitude and direction, but a scalar has magnitude only. They are represented by directed line segments with arrows. The length of the line segment represents the magnitude, while the arrow represents the direction.Â
This free online revision course is specially designed for students to revise important topics from A Level H2 Maths. The course content is presented in an easy to study format with 5 essential questions, core concepts and explanation videos for each question. Please download the worksheet and try the questions yourself! Have fun learning with us, consider joining our tuition classes or online courses.
2011 MJC P1 Q7 (i) (ii)
The lines ${{l}_{1}}$ and ${{l}_{2}}$ have equations $\mathbf{r}=\overrightarrow{OA}+\lambda \left( \begin{matrix} 1 \\-1 \\5 \\\end{matrix} \right)$ and $\mathbf{r}=\overrightarrow{OB}+\mu \left( \begin{matrix} 2 \\1 \\ 1 \\\end{matrix} \right)$, $\lambda $, $\mu \in \mathbb{R}$ respectively, where $\overrightarrow{OA}=\left( \begin{matrix} 1 \\ 2 \\ -1 \\\end{matrix} \right)$, $\overrightarrow{OB}=\left( \begin{matrix} 2 \\ 4 \\-5 \\\end{matrix} \right)$. The lines ${{l}_{1}}$ and ${{l}_{2}}$ intersect at point $C$.
(i)
Find the position of vector of $C$.
[3]
(i) Find the position of vector of $C$.
[3]
(ii)
Given that $\overrightarrow{AB}$ is perpendicular to ${{l}_{2}}$, find the equation of the reflection of ${{l}_{1}}$ in ${{l}_{2}}$.
[4]
(ii) Given that $\overrightarrow{AB}$ is perpendicular to ${{l}_{2}}$, find the equation of the reflection of ${{l}_{1}}$ in ${{l}_{2}}$.
[4]
Suggested Handwritten and Video Solutions
Share with your friends!
2012 CJC Promo Q10 (a)
The line ${{l}_{1}}$ passes through the points $A$ and $B$ with coordinates $(-1,-2,\,\,1)$ and $(0,\,\,1,\,\,5)$ respectively. The line ${{l}_{2}}$ has equation $x-1=\frac{y-2}{2}=z-3$. ${{l}_{1}}$ and ${{l}_{2}}$ intersect at the point $A$. Find
(i)
the vector equations of the lines ${{l}_{1}}$ and ${{l}_{2}}$,
[2]
(i) the vector equations of the lines ${{l}_{1}}$ and ${{l}_{2}}$,
[2]
(ii)
the acute angle between the lines ${{l}_{1}}$ and ${{l}_{2}}$,
[2]
(ii) the acute angle between the lines ${{l}_{1}}$ and ${{l}_{2}}$,
[2]
(iii)
the position vector of the foot of perpendicular, $F$, from $B$ to the line ${{l}_{2}}$,
[3]
(iii) the position vector of the foot of perpendicular, $F$, from $B$ to the line ${{l}_{2}}$,
[3]
(iv)
the equation of the line ${{l}_{3}}$ which is the mirror image of ${{l}_{1}}$ in ${{l}_{2}}$.
[3]
(iv) the equation of the line ${{l}_{3}}$ which is the mirror image of ${{l}_{1}}$ in ${{l}_{2}}$.
[3]
Suggested Handwritten and Video Solutions
Share with your friends!
ACJC Tutorial 15 Q9
The diagram shows a cuboid $OABCA’B’C’D’$ in which the lengths of $OA$,$OC$ and $OD’$ are $3a$,$2a$ and $a$ respectively, where $a\ne 0$. The point $O$ is taken as origin, with unit vectors $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ in the direction of $\overrightarrow{OA}$, $\overrightarrow{OC}$, $\overrightarrow{OD’}$ respectively. The points $P$ on $C’B’$ and $Q$ on $OD’$ are such that $C’P=\frac{1}{3}C’B’$ and $OQ=\frac{1}{3}OD’$ respectively.
(i)
Find the position vectors of $P$ and $Q$.
(i) Find the position vectors of $P$ and $Q$.
(ii)
Hence find the vector equation of line $PQ$.
(ii) Hence find the vector equation of line $PQ$.
(iii)
Determine whether the lines $PQ$ and $AC’$ are skew lines.
(iii) Determine whether the lines $PQ$ and $AC’$ are skew lines.
(iv)
Find the shortest distance from $M$, the midpoint of $AB’$ to the line $PQ$.
(iii) Find the shortest distance from $M$, the midpoint of $AB’$ to the line $PQ$.
(v)
Use vector product to find the area of triangle $PQM$.
(iv) Use vector product to find the area of triangle $PQM$.
Suggested Handwritten and Video Solutions
2020 TJC P2 Q4
(a)
Show that the triangle with vertices $A$, $B$ and $C$ is an isosceles right-angled triangle.
(a) Show that the triangle with vertices $A$, $B$ and $C$ is an isosceles right-angled triangle.
Â
(i)
It is given that $\mathbf{a}\times \mathbf{b}=\mathbf{0}$. Find $\mathbf{a}\cdot \mathbf{b}$ in terms of $\left| \mathbf{b} \right|$.
(i) It is given that $\mathbf{a}\times \mathbf{b}=\mathbf{0}$. Find $\mathbf{a}\cdot \mathbf{b}$ in terms of $\left| \mathbf{b} \right|$.
Â
It is now given that $\mathbf{a}\times \mathbf{b}\ne \mathbf{0}$. Point $P$ is the foot of the perpendicular from $A$ to the line $OB$ and the point $Q$ is the foot of the perpendicular from $B$ to the line $OA$. It is given that $AP=BQ$.
(ii)
Write down the lengths $AP$ and $BQ$ in terms of $\mathbf{a}$ and $\mathbf{b}$.
[2]
(ii) Write down the lengths $AP$ and $BQ$ in terms of $\mathbf{a}$ and $\mathbf{b}$.
[2]
Hence show that $\left| \mathbf{a} \right|=\left| \mathbf{b} \right|$.
[1]
(iii)
Given also that the angle between $\mathbf{a}$ and $\mathbf{a}-\mathbf{b}$ is $\phi $ radians, show that $\mathbf{a}\cdot \mathbf{b}=-{{\left| \mathbf{a} \right|}^{2}}\cos \,2\phi $.
[2]
(ii) Given also that the angle between $\mathbf{a}$ and $\mathbf{a}-\mathbf{b}$ is $\phi $ radians, show that $\mathbf{a}\cdot \mathbf{b}=-{{\left| \mathbf{a} \right|}^{2}}\cos \,2\phi $.
[2]
(b)
The pyramid $ORST$ has a triangular base $ORS$ and height $OT$. The position vectors of $R$ and $S$ are $-4\mathbf{i}+\mathbf{j}+3\mathbf{k}$ and $5\mathbf{i}+\mathbf{j}$ respectively. Find the possible coordinates of $T$ if the volume of the pyramid $ORST$ is $35$ units$^{3}$.
[5]
(b) The pyramid $ORST$ has a triangular base $ORS$ and height $OT$. The position vectors of $R$ and $S$ are $-4\mathbf{i}+\mathbf{j}+3\mathbf{k}$ and $5\mathbf{i}+\mathbf{j}$ respectively. Find the possible coordinates of $T$ if the volume of the pyramid $ORST$ is $35$ units$^{3}$.
[5]
[Volume of pyramid $=\frac{1}{3}\times $base area$\times $height]
Suggested Handwritten and Video Solutions
Share with your friends!
2019 ASRJC P1 Q9
The diagram above shows an object with $O$ at the centre of its rectangular base $ABCD$ where $AB=8$cm and $BC=4$cm. The top side of the object, $EFGH$ is a square with side $2$cm long and is parallel to the base. The centre of the top side is vertically above $O$ at a height $h$ cm.
(i)
Show that the equation of the line $BG$ may be expressed as $r=\left( \begin{matrix}4 \\-2 \\ 0 \\\end{matrix} \right)+t\left( \begin{matrix}-3 \\ 1 \\ h \\\end{matrix} \right)$, where $t$ is a parameter.
[1]
(i) Show that the equation of the line $BG$ may be expressed as $r=\left( \begin{matrix}4 \\-2 \\ 0 \\\end{matrix} \right)+t\left( \begin{matrix}-3 \\ 1 \\ h \\\end{matrix} \right)$, where $t$ is a parameter.
[1]
(ii)
Find the sine of the angle between the line $BG$ and the rectangular base $ABCD$ in terms of $h$.
[2]
(ii) Find the sine of the angle between the line $BG$ and the rectangular base $ABCD$ in terms of $h$.
[2]
It is given that $h=6$.
(iii)
Find the cartesian equation of the plane $BCFG$.
[3]
(iii) Find the cartesian equation of the plane $BCFG$.
[3]
(iv)
Find the shortest distance from the point $A$ to the plane $BCFG$.
[2]
(iv) Find the shortest distance from the point $A$ to the plane $BCFG$.
[2]
(v)
The line $l$, which passes through the point $A$, is parallel to the normal of plane $BCFG$. Given that, the line $l$ intersects the plane $BCFG$ at a point $M$, use your answer in part (iv) to find the shortest distance from point $M$ to the rectangular base $ABCD$.
[2]
(v) The line $l$, which passes through the point $A$, is parallel to the normal of plane $BCFG$. Given that, the line $l$ intersects the plane $BCFG$ at a point $M$, use your answer in part (iv) to find the shortest distance from point $M$ to the rectangular base $ABCD$.
[2]
Suggested Handwritten and Video Solutions
Share with your friends!
Download Vectors Worksheet
Learn more about our H2 Math Tuition
H2 Math Question Bank
Check out our question bank, where our students have access to thousands of H2 Math questions with video and handwritten solutions.
Share with your friends!