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Vectors, SPEC1 2023 VCAA 10

The position vector of a particle at time seconds is given by

, where .

  1. Write in the form , where .  (1 mark)
  1. Show that the Cartesian equation of the path of the particle is   (2 marks)
  1. The particle is at point when and at point when , where is a positive real constant.
  2. If the distance travelled along the curve from to is , find .   (1 mark)
  1. Find all values of for which the position vector of the particle, , is perpendicular to its velocity vector, .   (2 marks)
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a.   

b.   

c.   

d.   

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a. 


 

b.   

 
c. 


 

d.   

Vectors, SPEC2 2020 VCAA 4

A pilot is performing at an air show. The position of her aeroplane at time relative to a fixed origin is given by 

   ,

where    is a unit vector in a horizontal direction,   is a unit vector vertically up, displacement components are measured in metres and time is measured in seconds where  .

  1. Find the maximum speed of the aeroplane. Give your answer in .  (3 marks)

  2.  i. Use    to show that the cartesian equation of the path of the aeroplane is given by
  3.     .  (2 marks)

  4. ii. Sketch the path of the aeroplane on the axes provided below. Label the position of the aeroplane when  , using coordinates, and use an arrow to show the direction of motion of the aeroplane.  (3 marks)

  5.   

         
     

A friend of the pilot launches an experimental jet-powered drone to take photographs of the air show. The position of the drone at time relative to the fixed origin is given by  , where is in seconds and    is a unit vector in the same horizontal direction,   is a unit vector vertically up, and displacement components are measured in metres.

  1. Sketch the path of the drone on the axes provided in part b.ii. Using coordinates, label the points where the path of the drone crosses the path of the aeroplane, correct to the nearest metre.  (3 marks)

  2. Determine whether the drone will make contact with the aeroplane. Give reasons for your answer.  (3 marks)

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  2.  i.
  3. ii.
  4.  

     

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a.   
   
 
 

 

b. i.   


 


 

 

b. ii.   


 

 

c.   

 

 

 

d.   


 


  

Vectors, SPEC2 2020 VCAA 1

A particle moves in the    plane such that its position in terms of and metres at seconds is given by the parametric equations

where

  1. Find the distance, in metres, of the particle from the origin when  .   (2 marks)

  2.   i. Express    in terms of and, hence, find the equation of the tangent to the path of the particle at    seconds.   (3 marks)

  3.  ii. Find the velocity, , in , of the particle when  .   (2 marks)

  4. iii. Find the magnitude of the acceleration, in , when  .   (2 marks)

  5. Find the time, in seconds, when the particle first passes through the origin.   (1 mark)

  6. Express the distance, metres, travelled by the particle from    to    as a definite integral and find this distance correct to three decimal places.   (2 marks)

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  2.   i.
  3.  ii.
  4. iii.
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a.   

 

 

b.i.   

 

 

 


  

b.ii.  
 
 
   

 

b.iii. 

♦ Mean mark (b)(iii) 48%.
 
 

 

c.   

 

d.   
   
   

Vectors, SPEC1 2012 VCAA 9

The position of a particle at time    is given by

  1. Find the velocity of the particle at time     (1 mark)

  2. Find the speed of the particle at time    in the form  , where and are positive integers.   (2 marks)

  3. Show that at time     (2 marks)

  4. Find the angle in terms of , between the vector    and the vector    at time     (2 marks)

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a. 

 
   
   

 

b.   
   
 
   
   
   
   
   

♦ Mean mark part (c) 41%.

c.   
 
 
   
   
   

 

d.    

♦♦ Mean mark part (d) 32%.

 

 
 
 

Vectors, SPEC1 2013 VCAA 7

The position vector    of a particle moving relative to an origin at time seconds is given by

where the components are measured in metres.

  1. Show that the cartesian equation of the path of the particle is     (2 marks)

  2. Sketch the path of the particle on the axes below, labelling any asymptotes with their equations.   (2 marks)

     

        
     VCAA 2013 spec 7b
     

  3. Find the speed of the particle, in , when    (2 marks)

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  2.  
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a.  
 
 
 

  

 

b.   

♦ Mean mark part (b) 45%.

 

♦ Mean mark part (c) 39%.

c.   
   
   

 

 
 
 
 

Vectors, SPEC2 2018 VCAA 4

Two yachts, and , are competing in a race and their position vectors on a certain section of the race after time    hours are given by

where displacement components are measured in kilometres from a given reference buoy at origin .

  1. Find the cartesian equation of the path for each yacht.   (2 marks)

  2. Show that the two yachts will not collide if they follow these paths.   (2 marks)

  3. Find the coordinates of the point where the paths of the two yachts cross. Give your coordinates correct to three decimal places.   (2 marks)

One of the rules for the race is that the yachts are not allowed to be within 0.2 km of each other. If this occurs there is a time penalty for the yacht that is travelling faster.

  1. For what values of is yacht travelling faster than yacht ?   (2 marks)

  2. If yacht does not alter its course, for what period of time will yacht be within 0.2 km of yacht ? Give your answer in minutes, correct to one decimal place.   (2 marks)

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a. 

 

 

 

b. 

 


 

 

c.  
 
 
 
 
   

 

d.   
 
   
     
 
 
   

  
 

♦ Mean mark part (d) 41%.

 

 

e.   
   

 


 

♦ Mean mark part (e) 37%.

 

Vectors, SPEC1 2014 VCAA 2

The position vector of a particle at time    is given by

  1.  Show that the cartesian equation of the path followed by the particle is  .  (1 mark)

  2.  Sketch the path followed by the particle on the axes below, labelling all important features.  (2 marks)


 

 

  1.  Find the speed of the particle when  .  (2 marks)

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a.   
 
 

 

b.   

 

 

 

c.