Point on line satisfying distance or other condition

Edexcel C34 2014 June Q14

14 marks Standard +0.8

14. Relative to a fixed origin $O$, the line $l$ has vector equation $$\mathbf { r } = \left( \begin{array} { r } - 1 \\ - 4 \\ 6 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ 1 \\ - 1 \end{array} \right)$$ where $\lambda$ is a scalar parameter. Points $A$ and $B$ lie on the line $l$, where $A$ has coordinates ( $1 , a , 5$ ) and $B$ has coordinates ( $b , - 1,3$ ).

Find the value of the constant $a$ and the value of the constant $b$.
Find the vector $\overrightarrow { A B }$. The point $C$ has coordinates ( $4 , - 3,2$ )
Show that the size of the angle $C A B$ is $30 ^ { \circ }$
Find the exact area of the triangle $C A B$, giving your answer in the form $k \sqrt { 3 }$, where $k$ is a constant to be determined. The point $D$ lies on the line $l$ so that the area of the triangle $C A D$ is twice the area of the triangle $C A B$.
Find the coordinates of the two possible positions of $D$.

Edexcel C34 2019 June Q12

13 marks Standard +0.3

Relative to a fixed origin $O$,
the point $A$ has position vector $( 2 \mathbf { i } - 3 \mathbf { j } - 2 \mathbf { k } )$ the point $B$ has position vector $( 3 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } )$ the point $C$ has position vector ( $2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k }$ )

The line $l$ passes through the points $A$ and $B$.

Find the vector $\overrightarrow { A B }$.
Find a vector equation for the line $l$.
Show that the size of the angle $C A B$ is $62.8 ^ { \circ }$, to one decimal place.
Hence find the area of triangle $C A B$, giving your answer to 3 significant figures. The point $D$ lies on the line $l$. Given that the area of triangle $C A D$ is twice the area of triangle $C A B$,
find the two possible position vectors of point $D$.

Edexcel C4 2014 January Q8

15 marks Standard +0.3

8. With respect to a fixed origin $O$, the lines $l _ { 1 }$ and $l _ { 2 }$ are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 2 \\ - 3 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 2 \\ 1 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 2 \\ - 3 \\ 4 \end{array} \right) + \mu \left( \begin{array} { r } 5 \\ - 2 \\ 5 \end{array} \right)$$ where $\lambda$ and $\mu$ are scalar parameters.

Find, to the nearest $0.1 ^ { \circ }$, the acute angle between $l _ { 1 }$ and $l _ { 2 }$ The point $A$ has position vector $\left( \begin{array} { l } 0 \\ 1 \\ 6 \end{array} \right)$.
Show that $A$ lies on $l _ { 1 }$ The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $X$.
Write down the coordinates of $X$.
Find the exact value of the distance $A X$. The distinct points $B _ { 1 }$ and $B _ { 2 }$ both lie on the line $l _ { 2 }$ Given that $A X = X B _ { 1 } = X B _ { 2 }$
find the area of the triangle $A B _ { 1 } B _ { 2 }$ giving your answer to 3 significant figures. Given that the $x$ coordinate of $B _ { 1 }$ is positive,
find the exact coordinates of $B _ { 1 }$ and the exact coordinates of $B _ { 2 }$ \includegraphics[max width=\textwidth, alt={}, center]{245bbe52-3a14-4494-af17-7711caf79b22-28_96_59_2478_1834}

Edexcel C4 2013 June Q8

9 marks Standard +0.3

With respect to a fixed origin $O$, the line $l$ has equation

$$\mathbf { r } = \left( \begin{array} { c } 13 \\ 8 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ 2 \\ - 1 \end{array} \right) \text {, where } \lambda \text { is a scalar parameter. }$$ The point $A$ lies on $l$ and has coordinates ( $3 , - 2,6$ ).
The point $P$ has position vector ( $- p \mathbf { i } + 2 p \mathbf { k }$ ) relative to $O$, where $p$ is a constant.
Given that vector $\overrightarrow { P A }$ is perpendicular to $l$,

find the value of $p$. Given also that $B$ is a point on $l$ such that $\angle B P A = 45 ^ { \circ }$,
find the coordinates of the two possible positions of $B$. $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ Question 8 continued

OCR C4 2012 January Q7

6 marks Standard +0.3

7 The equation of a straight line $l$ is $$\mathbf { r } = \left( \begin{array} { l } 1 \\ 0 \\ 2 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ 0 \end{array} \right) .$$ $O$ is the origin.

Find the position vector of the point $P$ on $l$ such that $O P$ is perpendicular to $l$.
A point $Q$ on $l$ is such that the length of $O Q$ is 3 units. Find the two possible position vectors of $Q$. [3]

Edexcel Paper 2 2024 June Q7

5 marks Standard +0.8

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2ce10759-9ce6-47a1-b55d-d22082f88f55-16_330_654_246_751} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the straight line $l$.
Line $l$ passes through the points $A$ and $B$.
Relative to a fixed origin $O$

the point $A$ has position vector $2 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k }$
the point $B$ has position vector $5 \mathbf { i } + 6 \mathbf { j } + 8 \mathbf { k }$
1. Find $\overrightarrow { A B }$

Given that a point $P$ lies on $l$ such that $$| \overrightarrow { A P } | = 2 | \overrightarrow { B P } |$$

find the possible position vectors of $P$.

AQA C4 2012 June Q7

12 marks Standard +0.3

$\mathbf { 7 } \quad$ The line $l _ { 1 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } 0 \\ - 2 \\ q \end{array} \right] + \lambda \left[ \begin{array} { r } 2 \\ 0 \\ - 1 \end{array} \right]$, where $q$ is an integer. The line $l _ { 2 }$ has equation $\mathbf { r } = \left[ \begin{array} { l } 8 \\ 3 \\ 5 \end{array} \right] + \mu \left[ \begin{array} { l } 2 \\ 5 \\ 4 \end{array} \right]$. The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $P$.

Show that $q = 4$ and find the coordinates of $P$.
Show that $l _ { 1 }$ and $l _ { 2 }$ are perpendicular.
The point $A$ lies on the line $l _ { 1 }$ where $\lambda = 1$.
1. Find $A P ^ { 2 }$.
2. The point $B$ lies on the line $l _ { 2 }$ so that the right-angled triangle $A P B$ is isosceles. Find the coordinates of the two possible positions of $B$.

AQA C4 2008 June Q7

12 marks Standard +0.8

7 The coordinates of the points $A$ and $B$ are ( $3 , - 2,1$ ) and ( $5,3,0$ ) respectively. The line $l$ has equation $\mathbf { r } = \left[ \begin{array} { l } 5 \\ 3 \\ 0 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ 0 \\ - 3 \end{array} \right]$.

Find the distance between $A$ and $B$.
Find the acute angle between the lines $A B$ and $l$. Give your answer to the nearest degree.
The points $B$ and $C$ lie on $l$ such that the distance $A C$ is equal to the distance $A B$. Find the coordinates of $C$.

SPS SPS FM Pure 2022 June Q8

5 marks Standard +0.8

With respect to a fixed origin $O$, the line $l$ has equation $$\mathbf{r} = \begin{pmatrix} 13 \\ 8 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix}, \text{ where } \lambda \text{ is a scalar parameter.}$$ The point $A$ lies on $l$ and has coordinates $(3, -2, 6)$. The point $P$ has position vector $(-\mathbf{i} + 2\mathbf{k})$ relative to $O$. Given that vector $\overrightarrow{PA}$ is perpendicular to $l$, and that point $B$ is a point on $l$ such that $\angle BPA = 45°$, find the coordinates of the two possible positions of $B$. [5]