Foot of perpendicular from general external point to line

Edexcel C34 2019 January Q6

11 marks Standard +0.3

6. Relative to a fixed origin $O$, the points $A$, $B$ and $C$ have coordinates ( $2,1,9 ) , ( 5,2,7 )$ and $( 4 , - 3,3 )$ respectively. The line $l$ passes through the points $A$ and $B$.

Find a vector equation for the line $l$.
Find, in degrees, the acute angle between the line $I$ and the line $A C$. The point $D$ lies on the line $l$ such that angle $A C D$ is $90 ^ { \circ }$
Find the coordinates of $D$.
Find the exact area of triangle $A D C$, giving your answer as a fully simplified surd.

Edexcel C4 2011 January Q4

10 marks Moderate -0.3

Relative to a fixed origin $O$, the point $A$ has position vector $\mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$ and the point $B$ has position vector $- 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$. The points $A$ and $B$ lie on a straight line $l$.
1. Find $\overrightarrow { A B }$.
2. Find a vector equation of $l$.
The point $C$ has position vector $2 \mathbf { i } + p \mathbf { j } - 4 \mathbf { k }$ with respect to $O$, where $p$ is a constant. Given that $A C$ is perpendicular to $l$, find
the value of $p$,
the distance $A C$.

Edexcel C4 2013 January Q7

14 marks Standard +0.3

7. With respect to a fixed origin $O$, the lines $l _ { 1 }$ and $l _ { 2 }$ are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( 9 \mathbf { i } + 13 \mathbf { j } - 3 \mathbf { k } ) + \lambda ( \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } ) + \mu ( 2 \mathbf { i } + \mathbf { j } + \mathbf { k } ) \end{aligned}$$ where $\lambda$ and $\mu$ are scalar parameters.

Given that $l _ { 1 }$ and $l _ { 2 }$ meet, find the position vector of their point of intersection.
Find the acute angle between $l _ { 1 }$ and $l _ { 2 }$, giving your answer in degrees to 1 decimal place. Given that the point $A$ has position vector $4 \mathbf { i } + 16 \mathbf { j } - 3 \mathbf { k }$ and that the point $P$ lies on $l _ { 1 }$ such that $A P$ is perpendicular to $l _ { 1 }$,
find the exact coordinates of $P$.

Edexcel C4 2012 June Q8

10 marks Standard +0.3

Relative to a fixed origin $O$, the point $A$ has position vector $( 10 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } )$, and the point $B$ has position vector $( 8 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } )$.

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

Find the vector $\overrightarrow { A B }$.
Find a vector equation for the line $l$. The point $C$ has position vector $( 3 \mathbf { i } + 12 \mathbf { j } + 3 \mathbf { k } )$.
The point $P$ lies on $l$. Given that the vector $\overrightarrow { C P }$ is perpendicular to $l$,
find the position vector of the point $P$.

Edexcel C4 2018 June Q7

15 marks Standard +0.8

7. The point $A$ with coordinates ( $- 3,7,2$ ) lies on a line $l _ { 1 }$ The point $B$ also lies on the line $l _ { 1 }$ Given that $\quad \overrightarrow { A B } = \left( \begin{array} { r } 4 \\ - 6 \\ 2 \end{array} \right)$,

find the coordinates of point $B$. The point $P$ has coordinates ( $9,1,8$ )
Find the cosine of the angle $P A B$, giving your answer as a simplified surd.
Find the exact area of triangle $P A B$, giving your answer in its simplest form. The line $l _ { 2 }$ passes through the point $P$ and is parallel to the line $l _ { 1 }$
Find a vector equation for the line $l _ { 2 }$ The point $Q$ lies on the line $l _ { 2 }$ Given that the line segment $A P$ is perpendicular to the line segment $B Q$,
find the coordinates of the point $Q$.

AQA C4 2013 June Q6

14 marks Standard +0.8

6 The points $A , B$ and $C$ have coordinates $( 3 , - 2,4 ) , ( 1 , - 5,6 )$ and $( - 4,5 , - 1 )$ respectively. The line $l$ passes through $A$ and has equation $\mathbf { r } = \left[ \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right] + \lambda \left[ \begin{array} { r } 7 \\ - 7 \\ 5 \end{array} \right]$.

Show that the point $C$ lies on the line $l$.
Find a vector equation of the line that passes through points $A$ and $B$.
The point $D$ lies on the line through $A$ and $B$ such that the angle $C D A$ is a right angle. Find the coordinates of $D$.
The point $E$ lies on the line through $A$ and $B$ such that the area of triangle $A C E$ is three times the area of triangle $A C D$. Find the coordinates of the two possible positions of $E$.

AQA C4 2015 June Q6

12 marks Challenging +1.2

6 The points $A$ and $B$ have coordinates $( 3,2,10 )$ and $( 5 , - 2,4 )$ respectively.
The line $l$ passes through $A$ and has equation $\mathbf { r } = \left[ \begin{array} { r } 3 \\ 2 \\ 10 \end{array} \right] + \lambda \left[ \begin{array} { r } 3 \\ 1 \\ - 2 \end{array} \right]$.

Find the acute angle between $l$ and the line $A B$.
The point $C$ lies on $l$ such that angle $A B C$ is $90 ^ { \circ }$. \includegraphics[max width=\textwidth, alt={}, center]{fdd3905e-11f7-4b20-adfe-4c686018a221-12_360_339_762_852} Find the coordinates of $C$.
The point $D$ is such that $B D$ is parallel to $A C$ and angle $B C D$ is $90 ^ { \circ }$. The point $E$ lies on the line through $B$ and $D$ and is such that the length of $D E$ is half that of $A C$. Find the coordinates of the two possible positions of $E$.
[0pt] [4 marks]

AQA C4 2008 January Q9

11 marks Standard +0.3

9 The points $A$ and $B$ lie on the line $l _ { 1 }$ and have coordinates $( 2,5,1 )$ and $( 4,1 , - 2 )$ respectively.

1. Find the vector $\overrightarrow { A B }$.
2. Find a vector equation of the line $l _ { 1 }$, with parameter $\lambda$.
The line $l _ { 2 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } 1 \\ - 3 \\ - 1 \end{array} \right] + \mu \left[ \begin{array} { r } 1 \\ 0 \\ - 2 \end{array} \right]$.
1. Show that the point $P ( - 2 , - 3,5 )$ lies on $l _ { 2 }$.
2. The point $Q$ lies on $l _ { 1 }$ and is such that $P Q$ is perpendicular to $l _ { 2 }$. Find the coordinates of $Q$.

AQA C4 2009 January Q8

12 marks Standard +0.3

8 The points $A$ and $B$ have coordinates $( 2,1 , - 1 )$ and $( 3,1 , - 2 )$ respectively. The angle $O B A$ is $\theta$, where $O$ is the origin.

1. Find the vector $\overrightarrow { A B }$.
2. Show that $\cos \theta = \frac { 5 } { 2 \sqrt { 7 } }$.
The point $C$ is such that $\overrightarrow { O C } = 2 \overrightarrow { O B }$. The line $l$ is parallel to $\overrightarrow { A B }$ and passes through the point $C$. Find a vector equation of $l$.
The point $D$ lies on $l$ such that angle $O D C = 90 ^ { \circ }$. Find the coordinates of $D$.

AQA C4 2010 January Q8

11 marks Standard +0.3

8 The points $A , B$ and $C$ have coordinates $( 2 , - 1 , - 5 ) , ( 0,5 , - 9 )$ and $( 9,2,3 )$ respectively. The line $l$ has equation $\mathbf { r } = \left[ \begin{array} { r } 2 \\ - 1 \\ - 5 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right]$.

Verify that the point $B$ lies on the line $l$.
Find the vector $\overrightarrow { B C }$.
The point $D$ is such that $\overrightarrow { A D } = 2 \overrightarrow { B C }$.
1. Show that $D$ has coordinates $( 20 , - 7,19 )$.
2. The point $P$ lies on $l$ where $\lambda = p$. The line $P D$ is perpendicular to $l$. Find the value of $p$.

AQA C4 2005 June Q7

12 marks Standard +0.3

7 The points $A$ and $B$ have coordinates $( 1,4,2 )$ and $( 2 , - 1,3 )$ respectively.
The line $l$ has equation $\mathbf { r } = \left[ \begin{array} { r } 2 \\ - 1 \\ 3 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right]$.

Show that the distance between the points $A$ and $B$ is $3 \sqrt { 3 }$.
The line $A B$ makes an acute angle $\theta$ with $l$. Show that $\cos \theta = \frac { 7 } { 9 }$.
The point $P$ on the line $l$ is where $\lambda = p$.
1. Show that $$\overrightarrow { A P } \cdot \left[ \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right] = 7 + 3 p$$
2. Hence find the coordinates of the foot of the perpendicular from the point $A$ to the line $l$.

CAIE P3 2006 June Q10

12 marks Standard +0.3

The points $A$ and $B$ have position vectors, relative to the origin $O$, given by $$\overrightarrow{OA} = \begin{pmatrix} -1 \\ 3 \\ 5 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OB} = \begin{pmatrix} 3 \\ -1 \\ -4 \end{pmatrix}.$$ The line $l$ passes through $A$ and is parallel to $OB$. The point $N$ is the foot of the perpendicular from $B$ to $l$.

State a vector equation for the line $l$. [1]
Find the position vector of $N$ and show that $BN = 3$. [6]
Find the equation of the plane containing $A$, $B$ and $N$, giving your answer in the form $ax + by + cz = d$. [5]