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Vectors, EXT2 V1 2023 HSC 15b

On the triangular pyramid is the midpoint of is the midpoint of is the midpoint of is the midpoint of is the midpoint of and is the midpoint of .
 

Let    and  .

  1. Show that .  (1 mark)

  2. It can be shown that

  and

.   (Do NOT prove these.) 

  1. Prove that

 

  (3 marks)

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i.    
   
   
   
   

 

ii.   

.

 
   
   
   
   
   
   
   
   
   
   
   
   
   

♦♦ Mean mark (ii) 34%.

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 2021 HSC 12e

The diagram shows the pyramid    where    is a square. The diagonals of the square bisect each other at .
 

  1. Show that     (1 mark)

    Let be the point such that  .

  1. Using part (i), or otherwise, show that  .   (2 marks)

  2. Find the value of  λ  such that  λ   (1 mark)

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i.    


 

 
 
 
ii.    

 
 
 
iii. 
♦ Mean mark 48%.

 

λ

Vectors, EXT2 V1 2019 SPEC2 4

The base of a pyramid is the parallelogram    with vertices at points    and  . The apex (top) of the pyramid is located at  .

  1. Find the values of    and  .  (2 marks)

  2. Find the cosine of the angle between the vectors    and  .  (2 marks)

  3. Find the area of the base of the pyramid.  (2 marks)

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  3. ²
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i.   
   

 

 

ii.   

 
 

 

iii.   
 

  

  

  ²