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Vectors, EXT1 V1 2022 SPEC1 6b

is a semicircle of radius with equation  . is a point on the semicircle , as shown below.

  1. Express the vectors and in terms of  , , , and , where is a unit vector in the direction of the positive -axis and is a unit vector in the direction of the positive -axis.   (1 mark)

  2. Hence, using the vector scalar (dot) product, determine whether is perpendicular to .   (3 marks)

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i.   


 

ii.  
 
 

 

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i.   


 

ii.   
 
 

 

Vectors, EXT1 V1 2023 HSC 14c

  1. Given a non-zero vector  ,  it is known that the vector  is perpendicular to    and has the same magnitude. (Do NOT prove this.)
  2. Points and have position vectors    and  , respectively.
  3. Using the given information, or otherwise, show that the area of triangle    is  .  (3 marks)

  4. The point lies on the circle centred at    with radius  ,  such that    makes an angle of to the horizontal.

  5. The point lies on the circle centred at    with radius  ,  such that    makes an angle of to the horizontal.

  1. Note that    and  .
  2. Using part (i), or otherwise, find the values of , where  , that maximise the area of triangle .  (4 marks)

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i.   
     

♦♦♦ Mean mark (i) 18%.
 
   
   

 

 
   

 

ii.   
   
   
♦♦♦ Mean mark (ii) 8%.
 
 

 

  
   
   
   
   

 

 
   
   
   
   

 

 

Vectors, EXT1 V1 SM-Bank 5

Points and are on the number plane. The vector    is  .

Point is chosen so that the area of    is    square units and  .

Find all possible vectors  .  (4 marks)

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COMMENT: A diagram is highly recommended. Here, the possibility of 4 solutions is clearly shown.