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###### A-Level H2 Math | 5 Essential Questions

# Vectors

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**2011 MJC P1 Q7 (i) (ii)**

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**2012 CJC Promo Q10 (a)**

**Suggested Handwritten and Video Solutions**

**ACJC Tutorial 15 Q9**

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**2020 TJC P2 Q4**

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 $AP=BQ$.

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**2019 ASRJC P1 Q9**

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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

- Q1
- Q2
- Q3
- Q4
- Q5

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]

- (i)
- (ii)

- (i)
- (ii)

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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]

- (i)
- (ii)
- (iii)
- (iv)

- (i)
- (ii)
- (iii)
- (iv)

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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 $PQM$.

- (i)
- (ii)
- (iii)
- (iv)
- (v)

- (i)
- (ii)
- (iii)
- (iv)
- (v)

(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

(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]

- (a)(i)
- (a)(ii)
- (a) (iii)
- (b)

- (a) (i)
- (a) (ii)
- (a) (iii)
- (b)

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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]

- (i)
- (ii)
- (iii)
- (iv)
- (v)

- (i)
- (ii)
- (iii)
- (iv)
- (v)

- (i)
- (ii)
- (iii)
- (iv)
- (v)

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