Vector Equations of Lines (DP IB Maths: AI HL)

Exam Questions

4 hours29 questions
1a
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3 marks

The points straight A and straight B are given by straight A left parenthesis 4 comma space 2 comma negative 3 right parenthesis and straight B left parenthesis 0 comma space 5 comma space 1 right parenthesis.

Find a vector equation of the line straight L that passes through points straight A and straight B.

1b
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3 marks

Determine if the point straight C left parenthesis negative 1 comma space 3 comma space 2 right parenthesis does not lie on the line straight L.

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2
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5 marks

Find the vector equations of a line that is parallel to the vector bold italic a equals 3 bold i minus 4 bold j plus bold italic k and passes through the point straight X left parenthesis 3 comma negative 2 comma space 0 right parenthesis.

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3
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6 marks

Find the equation of the line that is perpendicular to the vector 4 bold italic i plus 5 bold italic j and passes through the point straight P left parenthesis 7 comma negative 1 right parenthesis, leaving your answer in the form a x plus b y plus c equals 0 comma where a comma space b and c element of straight integer numbers.

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4a
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2 marks

Consider the two lines l subscript 1 and l subscript 2 defined by the equations: 

l subscript 1 colon bold italic a equals open parentheses table row 4 row 1 row 6 end table close parentheses plus lambda open parentheses table row 1 row cell negative 3 end cell row cell negative 5 end cell end table close parentheses 

l subscript 2 colon bold italic b equals open parentheses table row 5 row cell negative 11 end cell row 10 end table close parentheses plus mu open parentheses table row cell negative 1 end cell row 6 row 2 end table close parentheses 

Find the scalar product of the direction vectors.

4b
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4 marks

Hence, find the angle, in radians, between the l subscript 1 and l subscript 2.

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5a
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2 marks

Consider the lines l subscript 1 and l subscript 2 defined by: 

l subscript 1 colon space open curly brackets table attributes columnalign left end attributes row cell x equals 3 minus mu space end cell row cell y equals negative 2 plus 5 mu end cell row cell z equals 4 plus 2 mu end cell end table close

l subscript 2 colon space bold r equals open parentheses table row 3 row cell negative 1 end cell row 0 end table close parentheses plus lambda open parentheses table row 4 row 2 row 2 end table close parentheses. 

Show that the lines are not parallel.

5b
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5 marks

Hence, show that the lines l subscript 1 and l subscript 2 do not intersect.

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6a
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2 marks

Consider the line l which can be defined by both bold italic r subscript bold 1 equals open parentheses table row t row cell negative 2 end cell row 5 end table close parentheses space plus space alpha open parentheses table row cell negative 5 end cell row 2 row 1 end table close parentheses and 

bold italic r subscript bold 2 equals open parentheses table row cell negative 3 end cell row 6 row 9 end table close parentheses plus beta open parentheses table row 15 row cell 3 k end cell row cell negative 3 end cell end table close parentheses.

Find the value of k.

6b
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4 marks

Find the value of t.

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7a
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2 marks

Consider the line l subscript 1, which can be represented by the equation bold italic r equals open parentheses table row 4 row 2 row cell negative 2 end cell end table close parentheses space plus lambda open parentheses table row cell negative 1 end cell row 4 row 3 end table close parentheses and l subscript 2, which can be represented by the equation bold italic s equals left parenthesis 3 minus mu right parenthesis i plus left parenthesis 1 minus mu right parenthesis j plus left parenthesis 5 plus 7 mu right parenthesis k.

Write down the equation for l subscript 2 in its vector form.

7b
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2 marks

Find vector product of the direction vectors of l subscript 1 and l subscript 2.

7c
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3 marks

Hence find the angle between l subscript 1 and l subscript 2.

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8a
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2 marks

The lines l subscript 1 and l subscript 2 can be defined by: 

l subscript 1 colon bold italic r equals open parentheses table row 2 row cell negative 5 end cell row 1 end table close parentheses plus alpha open parentheses table row 3 row 2 row k end table close parentheses 

l subscript 2 colon bold italic s equals open parentheses table row cell negative 3 end cell row cell negative 4 end cell row 2 end table close parentheses plus beta open parentheses table row cell negative 11 end cell row cell negative 3 end cell row 5 end table close parentheses 

Write down the parametric equations for l subscript 1.

8b
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7 marks

Given that l subscript 1and l subscript 2 intersect at point straight T, 

(i)
find the value of k. 

(ii)
determine the coordinates of the point of intersection, straight T.

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9a
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2 marks

Consider the triangle ABC. The points straight A, straight B and straight C have coordinates left parenthesis 4 comma space 0 comma negative 3 right parenthesis comma space left parenthesis 2 comma negative 2 comma negative 1 right parenthesis and left parenthesis 7 comma space 1 comma space 5 right parenthesis respectively.

M is the midpoint of open square brackets AB close square brackets. 

Find the coordinates of the midpoint M.

9b
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2 marks

Hence, find a vector equation of the line, l, that passes through points straight C and straight M.

9c
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3 marks

Show that the line l is perpendicular to [AB].

9d
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3 marks

Hence calculate the area of the triangle ABC.

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1a
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5 marks

Point A has coordinates open parentheses 7 comma space minus 1 comma space 20 close parentheses  and the line l  is defined by the equations:

l colon open curly brackets table attributes columnalign left end attributes row cell x equals 3 plus lambda end cell row cell y equals 2 lambda minus 1 end cell row cell z equals lambda end cell end table close

Point straight B lies on the line l such that open square brackets AB close square brackets is perpendicular to l.

Find the coordinates of point straight B.

1b
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2 marks

Hence find the shortest distance from A to the line l.

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2a
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1 mark

Find the vector equation of the line l subscript 1 with Parametric equations

l subscript 1 colon space open curly brackets table attributes columnalign left end attributes row cell x equals 4 lambda minus 3 end cell row cell y equals 2 plus 5 lambda end cell row cell z equals 4 lambda plus 3 end cell end table close 

2b
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4 marks

A second line l subscript 2  runs parallel to l subscript 1 and passes through the points straight X left parenthesis t space comma 2 comma negative 3 right parenthesis and straight Y space left parenthesis 23 comma 22 comma q right parenthesis.

Find the value of t and q .

2c
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1 mark

Hence write down the equation of line l subscript 2 in Parametric form.

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3
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6 marks

A line l passes through the points straight P left parenthesis 6 comma space 5 comma space minus 2 right parenthesis and straight Q open parentheses 2 x plus 2 comma space x minus 5 comma space x close parentheses and lies perpendicular to the vector 3 i plus 4 j minus k.

Find the vector equation of  l.

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4
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6 marks

Find the obtuse angle formed by the two lines l subscript 1 and l subscript 2 defined by the equations:

l subscript 1 colon open curly brackets table row cell x equals 4 minus 2 lambda end cell row cell y equals 1 plus 5 lambda end cell row cell z equals lambda minus 1 end cell end table close

l subscript 2 colon open curly brackets table row cell x equals 4 plus 3 mu end cell row cell y equals 18 plus mu end cell row cell z equals 6 plus 2 mu end cell end table close

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5a
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3 marks

Consider the two lines l subscript 1 and l subscript 2 as defined by:

l subscript 1 colon open curly brackets table row cell x equals 5 plus mu end cell row cell y equals 3 minus mu end cell row cell z equals 2 mu minus 8 end cell end table close

l subscript 2 colon r equals open parentheses table row cell negative 4 end cell row 3 row 1 end table close parentheses plus lambda open parentheses table row 2 row cell negative 5 end cell row 2 end table close parentheses

Find a vector that is perpendicular to both lines.

5b
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5 marks

Hence find the shortest distance between the two lines.

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6
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6 marks

Consider the lines  l subscript 1 and l subscript 2 defined by the equations:

 l subscript 1 colon open curly brackets table row cell x equals 2 plus 6 lambda end cell row cell y equals 2 plus q lambda end cell row cell z equals negative 8 minus 5 lambda end cell end table close

l subscript 2 colon r equals open parentheses table row cell negative 4 end cell row 5 row p end table close parentheses plus lambda open parentheses table row cell negative 24 end cell row 12 row 20 end table close parentheses 

Given that  l subscript 1 and l subscript 2 are identical, find the value of p and q.

 

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7a
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3 marks

Consider the two lines l subscript 1 and l subscript 2 defined by the equations:

l subscript 1 colon r equals open parentheses table row 4 row 4 row cell negative 3 end cell end table close parentheses plus lambda open parentheses table row 2 row 1 row cell negative 2 end cell end table close parentheses

l subscript 2 colon r equals open parentheses table row 3 row cell negative 1 end cell row cell negative 2 end cell end table close parentheses plus mu open parentheses table row cell negative 1 end cell row 1 row 4 end table close parentheses.

Show that the lines are not parallel and do not intersect.

7b
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4 marks

Calculate the exact value of the acute angle between the lines.

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8a
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2 marks

A helicopter is hovering in the sky at coordinates (4.5, 8, 2.7) relative to a helipad positioned on the ground at the origin, O.

The x direction is due east, the y direction is due north and the z direction is vertically upwards. The distances are measured in kilometres.

Write down the equation of a line the helicopter should travel along for it to travel directly to the helipad.

8b
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4 marks

Assuming the helicopter travels directly towards the helipad, but stops to hover at a point, P comma 0.54 km vertically above the ground, find

i)
the coordinates of the point P,
ii)
the distance the helicopter has left to travel on its final descent to the helipad, given that it continues along the most direct route.
8c
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5 marks

Assuming instead the helicopter travels directly towards a point, Q, 0.04 km vertically above the helipad, and then descends vertically downwards to the ground, find

i)
The coordinates of the point Q,
ii)
The component of the direction the helicopter actually travelled in that is perpendicular to the direction vector found in part (a),
iii)
The distance the helicopter has travelled from its starting position, including the vertical descent from Q to the helipad.

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9a
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3 marks

Consider the triangle ABC. The points A, B and C have coordinates open parentheses negative 6 comma space 3 comma space 13 close parentheses comma space open parentheses 4 comma space 5 comma space minus 8 close parentheses  and open parentheses 3 comma space minus 4 comma space t close parentheses  respectively.  A vector equation of the line that passes through point straight A and the midpoint of open square brackets BC close square brackets  is r equals open parentheses table row cell negative 6 end cell row 3 row 13 end table close parentheses plus lambda open parentheses table row 19 row cell negative 5 end cell row cell negative 27 end cell end table close parentheses 

Find the value of t.

9b
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3 marks

Find the vector equation of the line that passes through point B and the midpoint of [AC].

9c
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7 marks

The two lines intersect inside the triangle at point X.

Show that the area of AXC is  1 third the area of triangle ABC .

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10
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7 marks

In the magical kingdom of Cartesia, all positions are measured relative to the ancient stone of power known as the Origin. This reference system corresponds to the standard x comma space y comma space z coordinate system used in mathematics, as shown in the diagram below.

q10-_3-10_vector-equations-of-lines_hard_ib_aa_hl_math_dig

Prince Vector, son of the King Prime of Cartesia, needs to fly on his magical unicorn from the top of the Mystic Pedestal all the way to Cloud City, on an urgent rescue mission. 

The Mystic Pedestal is 14 kilometres west and 8 kilometres north of the Origin, and its top is one kilometre up from the level of the Origin. Cloud City is 11 kilometres east and 13 kilometres north of the Origin, and it is 11 kilometres up from the level of the Origin. 

Since there is not much time, the prince must fly directly from the top of the Mystic Pedestal to Cloud City. Unfortunately, the unicorn’s magic levels are low. In order for the unicorn to recharge it must pass within 12 kilometres of the Origin during the flight, and must do this before reaching the halfway point between the Mystic Pedestal and Cloud City. If the unicorn does not recharge before this point then it and the prince will crash into the barren wastes and the kingdom will perish. 

Using a vector method, determine whether or not the prince will reach Cloud City successfully. Use clear mathematical workings to justify your answer.

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1
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7 marks

The line l has equation r equals open parentheses table row 4 row 0 row 3 end table close parentheses plus lambda open parentheses table row cell negative 1 end cell row cell negative 2 end cell row 5 end table close parentheses and point A has coordinates open parentheses 3 comma space t comma space 2 close parentheses. Given that the shortest distance between point A and the line is fraction numerator square root of 645 over denominator 15 end fractionunits, find t , where t element of straight integer numbers.

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2a
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6 marks

A line l subscript 1 has the equation r subscript 1 equals open parentheses 2 plus lambda close parentheses straight i plus open parentheses 6 lambda minus 3 close parentheses straight j plus open parentheses 5 plus 2 lambda close parentheses k and intersects the line l subscript 2 with equation r subscript 2 equals 5 straight i plus open parentheses 7 minus 4 mu close parentheses straight j plus open parentheses negative 3 minus 7 mu close parentheses k at point P, when lambda equals 3.

A third line l subscript 3 runs parallel to l subscript 1 and also intersects l subscript 2 at point X open parentheses t comma space t minus 2 comma space minus 2 t close parentheses.  

Find the parametric equations of l subscript 3.

2b
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2 marks

Find the distance open vertical bar PX close vertical bar.

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3a
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4 marks

Consider the two intersecting lines l subscript 1 and l subscript 2 defined by the equations:

l subscript 1 colon r equals open parentheses table row 9 row 18 row 11 end table close parentheses plus lambda open parentheses table row cell negative 6 end cell row cell negative 3 end cell row k end table close parentheses

l subscript 2 colon open curly brackets table row cell x equals 2 mu minus 5 end cell row cell y equals negative 4 mu minus t end cell row cell z equals 3 mu plus 20 end cell end table close

Given that the angle between l subscript 1 and l subscript 2  is 1.281 radians, correct to 4 significant figures, find the value of k, where k element of straight integer numbers.

3b
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3 marks

Find the value of t, giving your answer correct to 3 significant figures.

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4
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8 marks

Consider the two lines l subscript 1 and l subscript 2 , where l subscript 1 passes through the points straight A left parenthesis 11 comma negative 2 comma space 3 right parenthesis  and straight B open parentheses 4 comma space 4 comma space minus 5 close parentheses  and l subscript 2  is defined by the Parametric equations:  

 l subscript 2 colon open curly brackets table row cell x equals 3 mu minus 7 end cell row cell 2 y equals 6 mu minus 9 end cell row cell z equals negative 4 mu minus 4 end cell end table close 

Find the shortest distance between the two lines.

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5a
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6 marks

Consider the line l subscript 1 as defined by the equation r subscript 1 equals open parentheses table row cell negative 2 end cell row 5 row cell negative 8 end cell end table close parentheses plus alpha open parentheses table row 2 row cell negative 1 end cell row 3 end table close parentheses. 

A point straight P left parenthesis r comma space t space comma negative r right parenthesis lies at a distance of square root of 405 units perpendicular from a point straight X open parentheses 17 comma space 15 comma negative 8 close parentheses  on l subscript 1.

Find all possible coordinates of P.

5b
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6 marks

Given that t greater than 0, write down the set of parametric equations that defines the line l subscript 2 that passes through points P and X.

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6a
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2 marks

A wheelchair ramp is required to provide access to a building with a door that is located 22 cm above ground level.  The maximum angle that a ramp must be from the horizontal is 4.8°.

Calculate the minimum horizontal distance that the ramp must extend out.

6b
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8 marks

The wheelchair ramp is supported by a steel frame.  A cross section of the ramp can be seen in the diagram below.  A metal strut joins M, the midpoint of [AC], to a point X on the line [AB]. [AB].XM=11.1 cm and straight M straight X with hat on top straight C=90°.  

q6a_3-10_vector-equations-of-lines_very-hard_ib_aa_hl_maths-diagram

Using the horizontal distance found in part (a) and assuming that point A is at the origin, use a vector method to calculate the length XB.

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7a
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4 marks

Some children are watching a canal boat navigating a system of locks. The boat starts at coordinates open parentheses negative 10 comma negative 2 comma negative 7 close parentheses  relative to the point at which the children are standing.

The xdirection is due east, the y direction is due north and the z direction is vertically upwards. All distances are measured in metres and the children are taken to be standing at the origin.

The boat travels with direction vector 1.5 straight i plus 2 straight j for 10 metres to get into the lock and then descends vertically downwards in the lock for 11 metres before continuing along the same direction vector as it was travelling along before entering the lock.

Find the coordinates of the entrance of the lock, given that the boat is now closer to the children.

7b
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2 marks

Find the equation of the line along which the boat is travelling after it leaves the lock

7c
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4 marks

On the next part of the journey at the point when the boat is closest to the children a child throws a flower to the boat driver. Given that the flower travels in a straight line and is caught by the boat driver, find the distance that the flower travelled.

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8a
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4 marks

Consider the tetrahedron ABCD, where straight A open parentheses 3 comma space 5 comma space 8 close parentheses, straight B open parentheses negative 2 comma space 3 comma space 2 close parentheses, straight C left parenthesis 5 comma space minus 1 comma space 3 right parenthesis   and straight D left parenthesis negative 3 comma space 0 comma space 1 right parenthesis . M is the midpoint of the line BC and point straight P lies along the line DM.

Given that the volume of the tetrahedron ABCP is  1 third of the volume of the tetrahedron ABCD, find the Vector equation of the line going through points A and P.

8b
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5 marks

 X is the midpoint of open square brackets AD close square brackets .

Find the coordinates of the point of intersection between the line found in part (a) and the line going through open square brackets MX close square brackets.

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9a
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5 marks

An adventure park structure is made out of steel rods arranged into a frame. As a part of the structure a red rod joins the coordinates open parentheses 2 comma negative 26 comma space 21 close parentheses to open parentheses negative 6 comma 14 comma negative 23 close parentheses and a blue rod joins open parentheses 16 comma negative 33 comma negative 46 close parentheses to open parentheses 6 comma space minus 18 comma negative 21 close parentheses.

Find the coordinates of the point where the red and blue rods meet each other.

9b
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4 marks

The red rod also meets a yellow rod which has the vector equation r equals open parentheses s minus 1 close parentheses straight i plus open parentheses s minus 29 close parentheses straight j plus open parentheses 8 s minus 3 close parentheses k. The point intersection of the red and blue rods and the red and yellow rods are joined by a taut rope.

Find the length of the rope.

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10a
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4 marks

A graphics designer joins the coordinates A open parentheses 1 comma 2 comma 3 close parentheses to B open parentheses 1 comma 0 comma 1 close parentheses and also plots the line l with parametric equations:

l colon open curly brackets table row cell x equals 3 minus 2 lambda end cell row cell y equals lambda minus 6 end cell row cell z equals 1 minus lambda end cell end table close

Find a Vector equation of the line joining the points A and B and show that it does not intersect the line l.

10b
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7 marks

Find the two possible coordinates of the point C on l such that the angle B A C is equal to  straight pi over 3 radians.  

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