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Vectors, EXT2 V1 2022 HSC 14a

  1. The two non-parallel vectors and satisfy    for some real numbers and .
  2. Show that  .  (2 marks)

  3. The two non-parallel vectors and satisfy    for some real numbers and .
  4. Using part (i), or otherwise, show that    and  . (1  mark)

The diagram below shows the tetrahedron with vertices and .

The point is defined by  , as shown in the diagram.

The point is the point of intersection of the straight lines and .
 
                       
 

  1. Using part (ii), or otherwise, determine the position of by showing that  .  (2 marks)

  2. The point is defined by  .
  3. Does lie on the line ? Justify your answer.  (2 marks)

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i.   
 

 


 

ii. 

 
 

 


Mean mark (ii) 56%.
iii.  
   

 

 
   
   
   


♦♦♦ Mean mark (iii) 25%.

 
 
 
 

 


 

iv. 


♦♦♦ Mean mark (iv) 23%.
 
 
 
 

 

Vectors, EXT2 V1 SM-Bank 21

  1. Find the equation of the vector line    that passes through    and  .  (1 mark)

  2. A sphere has centre    at    and a radius of    units.
    Find the points where the vector line    meets the sphere.  (3 marks)

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i.   


 

ii.   


 

 

Vectors, EXT2 V1 SM-Bank 15

Consider the two vector line equations

  1. Show that    and    intersect and determine the point of intersection . (2 marks)

  2. What is the acute angle between the vector lines, to the nearest minute.  (2 marks)

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i.   

 


  

 

ii.