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Vectors, EXT2 V1 2024 HSC 15a

Consider the three vectors  and , where is the origin and the points and are all different from each other and the origin.

The point is the point such that  .

  1. Show that lies on the line passing through and .   (1 mark)

  2. The point is the point such that  .
  3. Show that lies on the line passing through and , and lies between and .   (2 marks)

  4. The complex numbers and are all different and all have modulus 1.
  5. Using part (ii), or otherwise, show that  is never a cube root of .   (2 marks)

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i.   

 
   
   
   

 
.
 

ii.   

 
   
   
   

 

.
 

iii. 
 

Show Worked Solution

i.   

 
   
   
   

 
.
 

ii.   

 
   
   
   

 

.

♦ Mean mark (ii) 43%.

iii. 
 

♦♦♦ Mean mark (iii) 10%.

Vectors, EXT2 V1 SM-Bank 6

  1. What vector line equation, , corresponds to the Cartesian equation
  2.   (1 mark)

  3. Express     in Cartesian form where,
  4.   (1 mark)

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  1.  
  2.  
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i.   

COMMENT: Ensure you know the format for conversion from Cartesian to vector form (part i)!

 

ii.   

 

Vectors, EXT2 V1 SM-Bank 2

  1. Find values of  , ,   and    such that    is a vector equation of a line that passes through    and  .  (2 marks)

  2. Determine whether    is perpendicular to  .  (1 mark)

  3. Express    in Cartessian form.  (1 mark)

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  1. , or

     

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i.   

 
 

 

 
 

 

 

 
 

 

  


 

ii.   


 

iii.