A LevelMaths: Pure 3CIEExam Questions7. Vectors7.3 Further Vectors

Further Vectors (CIE A Level Maths: Pure 3)

Exam Questions

3 hours34 questions
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1a
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The equation of a line is given in vector form as follows bold r equals 2 bold i plus 6 bold j plus t open parentheses 7 bold i plus 3 bold j close parentheses. space 


Explain briefly what the vectors  open parentheses space 2 bold i plus 6 bold j space space close parenthesesand  open parentheses 7 bold i plus 3 bold j close parentheses  in the equation above tell us, respectively, about the line.

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1b
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Write down, in vector form, the equation of the line passing through space open parentheses 1 comma space 4 close parentheses in the same direction as left parenthesis bold i + 3 bold j right parenthesis straight.

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2
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A line passes through the two points  A open parentheses 3 comma space minus 2 close parentheses  and space B open parentheses 5 comma space 7 close parentheses.

i)
Write down the position vectors  space stack O A space space with rightwards arrow on topand space stack O B space space with rightwards arrow on top of the two points.
ii)
Use the vector relation  stack A B with rightwards arrow on top equals stack O B with rightwards arrow on top minus stack O A with rightwards arrow on top  to find vector stack A B with rightwards arrow on top.
iii)
Use your answers from (i) and (ii) to write down an equation of the line in vector form.

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3a
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Write down in vector form the equation of the line through the point  open parentheses 1 comma space 3 comma space minus 6 close parentheses  in the direction  4 bold i minus bold j plus 2 bold k.

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3b
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By first calculating the vectors  stack O A with rightwards arrow on top comma space space stack O B with rightwards arrow on top  and  stack A B with rightwards arrow on top,  find a vector equation of the line passing through the two points  A open parentheses 1 comma space 3 comma space 1 space close parentheses and space B open parentheses 3 comma space 4 comma space minus 3 close parentheses

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4
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A and B are the points on the line bold space bold r equals open parentheses table row 2 row cell negative 1 end cell row 3 end table close parentheses space plus t open parentheses table row 0 row 4 row cell negative 3 end cell end table close parentheseswith space space t equals 1 space and space t equals 3 space space respectively.  

i)
Find the position vectors stack space O A with rightwards arrow on top spaceand stack O B with rightwards arrow on top.

ii)
By first finding the vector  stack A B with rightwards arrow on top,  calculate the modulus  open vertical bar stack A B with rightwards arrow on top close vertical bar.

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5a
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The coordinates of three points are  A open parentheses 2 comma space minus 1 comma space 3 close parentheses,  space B open parentheses 1 comma space minus 2 comma space 5 space space close parenthesesand  C open parentheses 6 comma space 3 comma space 1 close parentheses.

Find  stack A B with rightwards arrow on top  and  stack A C with rightwards arrow on top, and calculate  open vertical bar stack A B with rightwards arrow on top close vertical bar  and  open vertical bar stack A C with rightwards arrow on top close vertical bar .

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5b
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For two vectors space bold a equals a subscript 1 bold i plus a subscript 2 bold j plus a subscript 3 bold k  and space bold b equals straight b subscript 1 bold i plus b subscript 2 bold j plus b subscript 3 bold k,  the scalar product  bold a bold. bold b  can be calculated using the formula

         bold a bold. bold b equals straight a subscript 1 straight b subscript 1 plus straight a subscript 2 straight b subscript 2 plus straight a subscript 3 straight b subscript 3

Calculate the scalar product  stack A B. with rightwards arrow on top stack A C with rightwards arrow on top.

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5c
Sme Calculator2 marks

A defining property of the scalar product of two vectors  bold a  and  bold b  is

      bold a bold. bold b equals stretchy vertical line a stretchy vertical line stretchy vertical line b stretchy vertical line cos space theta

where theta is the angle between the two vectors.

Using this relationship, along with your answers to parts (a) and (b), find the angle between the vectors  stack A B space with rightwards arrow on top and  stack A C with rightwards arrow on top.

Give your answer in degrees correct to 1 decimal place.

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6a
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For two vectors  bold a  and bold space bold b,  the definition of the scalar product  bold a bold. bold b  is

         


Write down  cos space open parentheses 90 degree close parentheses ,  and explain why this value of cosine shows that if  bold a  and  bold b  are perpendicular then  bold a bold. bold b equals 0.

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6b
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For  bold a equals a subscript 1 bold i plus a subscript 2 bold j plus a subscript 3 bold k  and space bold b equals b subscript 1 bold i plus b subscript 2 bold j plus b subscript 3 bold k,  the scalar product can be calculated using the formula

         bold a bold. bold b equals straight a subscript 1 straight b subscript 1 plus straight a subscript 2 straight b subscript 2 plus straight a subscript 3 straight b subscript 3  

Additionally it can be shown that if space bold a bold. bold b equals 0 space space then  bold a  and  bold b are perpendicular.


Use the above results to prove that the vectors stack A B with rightwards arrow on top space equals space minus 3 bold i plus bold j plus 5 bold k and space stack A C with rightwards arrow on top equals bold i minus 7 bold j plus 2 bold k bold space bold spaceare perpendicular.

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7a
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Two lines l and m have equations space bold r equals bold i plus bold j plus 3 bold k plus s open parentheses bold i minus bold j plus bold k space space close parentheses and bold r equals bold i minus 2 bold j plus 2 bold k plus t open parentheses bold 2 bold i plus bold j plus 3 bold k close parentheses  respectively.  The two lines intersect at a point P.

Explain why, at point P, the values of the parameters s and t must satisfy the vector equation

         open parentheses table row cell 1 plus s end cell row cell 1 minus s end cell row cell 3 plus s end cell end table close parentheses equals open parentheses table row cell 1 plus 2 t end cell row cell negative 2 plus t end cell row cell 2 plus 3 t end cell end table close parentheses

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7b
Sme Calculator3 marks

Solve the simultaneous equations 1 plus s equals 1 plus 2 t space spaceand  space 1 minus s equals negative 2 plus t comma space spaceand show that the values found for s and space t spacealso satisfy the equation  3 plus s equals 2 plus 3 t.

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7c
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Hence, find the coordinates of the point P.

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8a
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Two lines l and m have equations bold space bold r equals 6 bold i minus 2 bold j plus bold k plus s open parentheses bold i plus bold j minus bold k close parentheses  and space bold space bold r equals 3 bold i plus 3 bold j minus 2 bold k plus straight t open parentheses bold 3 bold i minus bold j plus bold k space close parentheses respectively.

Show that if the two lines intersect, then at the point of intersection the parameters s and t must satisfy the vector equation

         open parentheses table row cell 6 plus s end cell row cell negative 2 plus s end cell row cell 1 minus s end cell end table close parentheses space equals space open parentheses table row cell 3 plus 3 t end cell row cell 3 minus t space end cell row cell negative 2 plus t end cell end table close parentheses

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8b
Sme Calculator3 marks

Solve the simultaneous equations  6 plus s equals 3 plus 3 t space and  negative 2 plus s equals 3 minus t comma  and show that the values found for s and space t spacedo not also satisfy the equation space space 1 minus s equals negative 2 plus t.

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8c
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What do the results of part (b) tell you about the lines l and m?

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9a
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The line l has equation  space bold r equals open parentheses table row 2 row cell negative 3 end cell row 4 end table close parentheses plus s space open parentheses table row 1 row 2 row cell negative 1 end cell end table close parentheses .

Point N is the point on line space l such that the line connecting N and the point  P open parentheses 1 comma space 7 comma space 5 close parentheses is perpendicular to l.

Explain why  stack O N with rightwards arrow on top equals open parentheses 2 plus s space minus 3 plus 2 s space 4 minus s space space close parentheses for some value of the parameter  s comma  and use that 

expression for stack space space O N with rightwards arrow on top  to show that the vector  stack N P with rightwards arrow on top  is given in terms of  s  by

stack N P with rightwards arrow on top equals open parentheses negative 1 minus s space 10 minus 2 s space 1 plus s close parentheses

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9b
Sme Calculator2 marks

Using the properties of the scalar product, explain why the value of  s  corresponding to point  N  must satisfy the equation

open parentheses negative 1 minus s space 10 minus 2 s space 1 plus s close parentheses. open parentheses space 1 space 2 space minus 1 space close parentheses equals 0

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9c
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By solving the equation in part (b), determine the coordinates of point  N.

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9d
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Given that the shortest distance from a point to a line is the perpendicular distance,  find the shortest distance from point P to the line l.  You should give your answer as an exact value.

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1a
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Find the equation of each of these lines in vector form.

The line joining left parenthesis 4 comma 1 right parenthesis to left parenthesis 7 comma 6 right parenthesis.

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1b
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The line passing through left parenthesis 2 comma 5 right parenthesis in the same direction as 2 bold i plus bold j.

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2a
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Find the equation of each of these lines in vector form.

(i)
Through left parenthesis 3 comma 1 comma negative 2 right parenthesis in the direction open parentheses table row 0 row 1 row 5 end table close parentheses
(ii)
Through left parenthesis 2 comma 0 comma 7 right parenthesis and left parenthesis 5 comma 2 comma negative 3 right parenthesis. 

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2b
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Show that the point left parenthesis negative 4 comma negative 4 comma 27 right parenthesis lies on the line found in part (a) (ii).

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3
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A and B are the points on the line bold italic r equals open parentheses table row 3 row cell negative 2 end cell row 5 end table close parentheses plus t open parentheses table row 1 row 3 row cell negative 1 end cell end table close parentheses   with t equals 4 and t equals 7 respectively. 

(i)
Find the position vectors stack O A with rightwards arrow on top and stack O B with rightwards arrow on top.
(ii)
Find open vertical bar stack A B with rightwards arrow on top close vertical bar.

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4a
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The coordinates of three points are A left parenthesis 1 comma 0 comma negative 4 right parenthesis comma space space B left parenthesis 3 comma negative 1 comma 6 right parenthesis space spaceand  C left parenthesis negative 2 comma 5 comma 7 right parenthesis.

Find stack A B with rightwards arrow on top and  stack A C with rightwards arrow on top.

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4b
Sme Calculator3 marks

Calculate open vertical bar stack A B with rightwards arrow on top close vertical bar and  open vertical bar stack A C with rightwards arrow on top close vertical bar,  and the scalar product  stack A B space with rightwards arrow on top times space stack A C with rightwards arrow on top.

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4c
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Hence, find the angle between the vectors stack A B with rightwards arrow on top and stack A C with rightwards arrow on top. 
Give your answer in degrees correct to 1 decimal place.

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5
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The vertices of triangle A B C are the points with coordinates A left parenthesis 2 comma 4 comma negative 3 right parenthesis comma space space B left parenthesis 0 comma 2 comma 1 right parenthesis and  C left parenthesis negative 4 comma negative 2 comma negative 3 right parenthesis.

(i)
Show that stack B A with rightwards arrow on top times stack B C with rightwards arrow on top equals 0.
(ii)
What does the result of part (i) tell you about the triangle A B C?

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6
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Two lines l and m have equations  bold r equals negative 3 bold i minus 6 bold j plus 8 bold k plus s left parenthesis bold i plus bold j minus bold k right parenthesis and  bold italic r equals 8 bold italic i minus 10 bold italic j plus 6 bold italic k plus t left parenthesis 2 bold i minus 3 bold j plus bold k right parenthesis respectively.

Show that the two lines meet and find the position vector of the point of intersection.

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7
Sme Calculator4 marks

Two lines l and m have equations bold r equals 8 bold i minus 5 bold j minus 6 bold k plus straight s left parenthesis negative bold i plus 2 bold j plus bold k right parenthesis and bold r equals 2 bold i plus 2 bold j minus 2 bold k plus straight t left parenthesis 2 bold i plus bold j plus bold k right parenthesis respectively.

(i)
Show that the two lines do not intersect.
(ii)
Are the two lines skew?  Be sure to justify your answer.

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8a
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The line l has equation r equals open parentheses table row 1 row cell negative 4 end cell row 10 end table close parentheses plus s open parentheses table row 1 row cell negative 1 end cell row 3 end table close parentheses.

Point N is the point on line l such that the line connecting N and the point P left parenthesis 3 comma 4 comma 1 right parenthesis is perpendicular to l.

Find the position vector, stack O N with rightwards arrow on top, of point N.

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8b
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Hence find the shortest distance from point P to the line l.

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9a
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The following diagram depicts a large bank vault, the walls, floor and ceiling of which are all perfectly rectangular.

q7a-7-3-medium-cie-a-level-maths-pure-3

The corner O is taken to be the origin of the coordinate system, and the units on the diagram are all given in metres.

 C E is the diagonal running across the room from corner C to corner E.

By first finding the vector ,  write a vector equation of the line that passes through  and .

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9b
Sme Calculator4 marks

A motion detector is located at point O, and is set to sound an alarm if anything moves within 5 metres of it.

If a small insect flies in a straight line from point C to point E, will the alarm be triggered?  You must provide clear mathematical workings to justify your answer.

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1a
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Find the equation of each of these lines in vector form.

The line joining left parenthesis 3 comma negative 2 right parenthesis to left parenthesis negative 1 comma 5 right parenthesis.

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1b
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The line passing through  left parenthesis 3 comma 1 right parenthesis parallel to 4 bold i minus 12 bold j.

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2a
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Find the equation of each of these lines in vector form.

(i)
Through left parenthesis 2 comma negative 7 comma 9 right parenthesis in the same direction as 6 bold i plus 3 bold j minus 9 bold k.
(ii)
Through left parenthesis 4 comma negative 3 comma 1 right parenthesis and left parenthesis negative 8 comma 3 comma negative 5 right parenthesis. 

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2b
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Show that the point left parenthesis 10 comma negative 6 comma 5 right parenthesis does not lie on the line found in part (a) (ii).

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3
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A and B are the points on the line r equals open parentheses table row 1 row cell negative 4 end cell row 3 end table close parentheses plus t open parentheses table row 2 row a row cell negative 1 end cell end table close parentheses with t equals negative 1 and t equals 2 respectively.  Given that the distance from A to B is 9 units, find the possible values of a.

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4a
Sme Calculator2 marks

The coordinates of three points are A left parenthesis 2 comma negative 1 comma 5 right parenthesis comma space space B left parenthesis 4 comma 1 comma 1 right parenthesis and  C left parenthesis negative 2 comma negative 5 comma 9 right parenthesis.

Find stack B A with rightwards arrow on top and stack B C with rightwards arrow on top.

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4b
Sme Calculator3 marks

By considering the scalar product stack B A with rightwards arrow on top times stack B C with rightwards arrow on top, or otherwise, calculate the angle between stack B A with rightwards arrow on top and  stack B C with rightwards arrow on top.  Give your answer in degrees, accurate to 1 decimal place.

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5
Sme Calculator5 marks

The vertices of triangle A B C are the points with coordinates A left parenthesis 5 comma negative 3 comma 0 right parenthesis comma space space B left parenthesis negative 2 comma 0 comma negative 1 right parenthesis and  C left parenthesis 1 comma 2 comma 1 right parenthesis.

(i)
Calculate the scalar products stack A B with rightwards arrow on top times stack A C with rightwards arrow on top comma space space space stack B A with rightwards arrow on top times stack B C with rightwards arrow on top and  stack C A with rightwards arrow on top times stack C B. with rightwards arrow on top
(ii)
What does the result of part (i) tell you about the triangle A B C?

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6
Sme Calculator10 marks

Determine whether each of the following pairs of lines intersect, are parallel, or are skew.  If the lines intersect, find the coordinates of the point of intersection.

(i)
bold italic r equals negative 7 bold italic i minus bold italic j plus 6 bold italic k plus s left parenthesis bold italic i plus bold italic j minus bold italic k right parenthesis   and    bold italic r equals 2 bold italic i plus 2 bold italic j plus 11 bold italic k plus t left parenthesis 2 bold italic i minus bold italic j plus 5 bold italic k right parenthesis
(ii)
bold italic r equals 3 bold italic i minus 3 bold italic j plus bold italic k plus s left parenthesis bold italic i minus 2 bold italic j plus bold italic k right parenthesis     and    bold italic r equals negative bold italic i minus bold italic j plus 5 bold italic k plus t left parenthesis 2 bold italic i plus 2 bold italic j minus 3 bold italic k right parenthesis.

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7
Sme Calculator5 marks

Find the perpendicular distance of the point P left parenthesis 6 comma 0 comma 3 right parenthesis  to the line bold italic r equals negative bold italic i plus 5 bold italic j plus 2 bold italic k plus s left parenthesis 3 bold italic i minus 6 bold italic j minus 2 bold italic k right parenthesis.

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8
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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 y comma z coordinate system used in mathematics, as shown in the diagram below:

q9-7-3-medium-cie-a-level-maths-pure-3

Prince Vector, son of 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 to save the kingdom from certain doom. 

The Mystic Pedestal is 14 miles west and 8 miles north of the Origin, and its top is one mile up from the level of the Origin.  Cloud City is 11 miles east and 13 miles north of the Origin, and it is 11 miles 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, his unicorn’s magic levels are low.  In order for the unicorn to recharge it must pass within 12 miles 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.

After the prince departs, his sister Hypatia (a keen mathematics student and heiress to the throne) remembers that there is a vector method that can be used to determine whether or not her brother’s mission will succeed.

Determine whether or not the prince will reach Cloud City successfully, using clear mathematical workings to justify your answer.

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1
Sme Calculator6 marks

Given that the coordinates of A and B are  open parentheses negative 1 comma space 7 close parentheses  and  open parentheses space minus 7 comma space minus 5 space close parenthesesrespectively, find the equation of each of the following lines in vector form.

i)
The line joining  open parentheses 5 comma space minus 6 space close parentheses to the midpoint of  A B.
ii)
The line passing through B, parallel to the line in part (i).

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2a
Sme Calculator5 marks

Given that the coordinates of A comma space B and space C spaceare  open parentheses 6 comma space 1 comma space minus 3 close parentheses comma space space open parentheses negative 2 comma space minus 7 comma space 5 space close parentheses and  open parentheses 3 comma space 12 comma space minus 9 close parentheses respectively, find the equation of each of the following lines in vector form.

i)
The line through A and B.

ii)
The line through B, parallel to  stack O C with rightwards arrow on top.

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2b
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Determine whether the point  open parentheses space 1 comma space 5 comma space minus 4 space space close parentheses lies on either of the lines found in part (a).

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3
Sme Calculator6 marks

The coordinates of three points are  A open parentheses negative 2 comma space 3 comma space minus 5 close parentheses,  B open parentheses 2 comma space minus 5 comma space minus 9 close parentheses  and  C open parentheses negative 10 comma space minus 1 comma space minus 1 close parentheses.  

The point M is the midpoint of A B, and the point N lies on B C.

Given that open vertical bar stack B N with rightwards arrow on top close vertical bar equals 3 open vertical bar stack N C with rightwards arrow on top close vertical bar, find the equation of the line through points M and N in vector form.

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4
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The coordinates of three points are  A open parentheses negative 3 comma space 4 comma space 0 close parentheses comma space space B open parentheses 2 comma space minus 2 comma space minus 3 close parentheses space space and space space C open parentheses negative 1 comma space minus 1 comma space minus 4 close parentheses.

Calculate the angle between space stack C A space space with rightwards arrow on top and space stack C B with rightwards arrow on top.  Give your answer in degrees, accurate to 1 decimal place.

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5
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The vertices of triangle space A B C spaceare the points with coordinates space A open parentheses negative 2 comma space 5 comma space 4 close parentheses comma space space B open parentheses 3 comma space 1 comma space 0 close parentheses space space and space space C open parentheses negative 1 comma space minus 3 comma space minus 1 close parentheses.

Use a vector method to prove that A B C is a right-angled triangle.

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6
Sme Calculator14 marks

Determine whether each of the following pairs of lines intersect, are parallel, or are skew.  For any lines that intersect, determine the point(s) of intersection.

i)
bold r equals 3 bold i minus 5 bold j plus bold k plus straight s open parentheses bold i plus 3 bold j minus 2 bold k space close parentheses and  italic r equals italic i plus 2 italic j plus 2 italic k and t open parentheses italic i minus sign italic j minus sign italic k close parentheses.

ii)
bold r equals bold i minus 2 bold j plus 3 bold k plus straight s open parentheses bold 2 bold i plus 6 bold j minus 2 bold k space close parentheses and  bold r equals 4 bold i bold space plus 7 bold j plus straight t open parentheses negative bold 5 bold i bold space minus 15 bold j plus 5 bold k space close parentheses

iii)
 bold r equals negative bold i minus 4 bold j minus 4 bold k plus straight s open parentheses bold 2 bold i minus 2 bold j minus 4 bold k space close parenthesesand  bold r equals negative 4 bold i bold space minus bold j minus 2 bold k space plus straight t open parentheses 3 bold i bold space minus 3 bold j plus 6 bold k space close parentheses

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7
Sme Calculator6 marks

Find the coordinates of the point on the line bold r equals 2 bold i minus 12 bold j plus 3 bold k plus s open parentheses bold i minus 6 bold j plus 4 bold k close parentheses space that is closest to the point  P open parentheses 2 comma space 3 comma space minus 1 close parentheses comma  and hence determine the minimum distance from point P to the line.

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8
Sme Calculator11 marks

The following diagram depicts imaginary lines connecting five points in space:

q8-vhard-7-3-furthervectors-cie-maths-pure-

Points A comma space B comma space C and D are the locations, respectively, of the stars Arccirclus, Betacarotjuse, α-Capella and Denomineb.  Point S is the location of the Stellamortis battle station, a planet-killing atrocity being built by the evil Galactic Imperium.  Coordinates are given relative to an origin point in accordance with the standard x comma space y comma space z coordinate system, and the units for all coordinates are parsecs.

The forces of the Star Rebellion are prepared to launch a strike to destroy the battle station, but they are unsure of its exact location.  According to data recovered from a smuggled droid, however, the following facts are known about the location of point S:

  • Point S is in the First Octant of the galaxy, where x comma space y and z coordinates are all positive.
  • The distance from point C to point S is exactly 45 square root of 2 parsecs.
  • Points B comma space C comma space D and S form the base of a pyramid, with its apex at point A.
  • The point on B D closest to point A is also the point where the two diagonals of the pyramid’s base intersect.

As the rebellion’s Chief Mathematician, it is your job to use the information provided to find the exact coordinates of point space S.  The fate of the galaxy is in your mathematical hands!

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