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Vectors, SPEC1 2021 VCAA 9

Let    and    be the position vectors relative to a fixed point of particle and particle respectively for  , where is a positive real constant.

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

    2. Show that the cartesian equation of the path of particle in the first quadrant can be written as  .   (1 mark)

    1. Show that the particles and will collide.   (1 mark)

    2. Hence, find the coordinates of the point of collision of the two particles.   (1 mark)

    1. Show that  .   (2 marks)


    2.    

      Hence, find the area bounded by the graph of  ,  the -axis and the lines    and  ,  as shown in the diagram above. Give your answer in the form  , where and are positive integers.   (2 marks)

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a.i.   

 

 

a.ii. 

♦ Mean mark part (a)(ii) 41%.

 

 
 

 

b.i.   

 

 

 

 

b.ii.   

 
♦ Mean mark part (c)(i) 37%.

 

c.i.  

♦ Mean mark part (c)(ii) 45%.

 

c.ii.   
   
   
   
   
   
   
   

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, SPEC1 2019 VCAA 4

The position vectors of two particles    and    at time  seconds after they have started moving are given by    and    respectively, where    is a real constant and  .

Find the value of    if the particles collide after they have started moving.   (3 marks)

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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%.