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Vectors, EXT2 V1 2024 HSC 16a

Consider the function  , where  . The value of has been chosen so that a circle can be drawn, centred at the origin, which has exactly two points of intersection with the graph of the function and so that the circle is never above the graph of the function. The point    is the point of intersection in the first quadrant, so    and  ,  as shown in the diagram.

The vector joining the origin to the point is perpendicular to the tangent to the graph of the function at that point. (Do NOT prove this.)

Show that  .   (4 marks)

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♦♦♦ Mean mark 8%.

 
   

 

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Vectors, EXT2 V1 2023 HSC 10 MC

Consider any three-dimensional vectors    and    that satisfy these three conditions

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Which of the following statements about the vectors is true?

  1. Two of and could be unit vectors.
  2. The points and could lie on a sphere centred at .
  3. For any three-dimensional vector , vectors and can be found so that and satisfy these three conditions.
  4. and satisfying the conditions, and such that and are positive real numbers and  .
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♦♦♦ Mean mark 15%.

Vectors, EXT2 V1 2023 HSC 15c

A curve spirals 3 times around the sphere centred at the origin and with radius 3, as shown.

A particle is initially at the point and moves along the curve on the surface of the sphere, ending at the point .
 

By using the diagram below, which shows the graphs of the functions    and  , and considering the graph  , give a possible set of parametric equations that describe the curve .  (3 marks)
 

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♦♦♦ Mean mark 22%.

Vectors, EXT2 V1 EQ-Bank 11

Point is the circumcentre of triangle which is the centre of the circle that passes through each of its vertices.

Point is the centroid of triangle where the bisectors of each angle within the triangle intersect.

Point is such that .
 

Prove that     (5 marks)

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Vectors, EXT2 V1 2021 HSC 16a

  1. The point    lies on the sphere of radius 1 centred at the origin .
  2. Using the position vector of , and the triangle inequality, or otherwise, show that .  (2 marks)

  3. Given the vectors    and  , show that 
  4.    .  (3 marks)

  5. As in part (i), the point    lies on the sphere of radius 1 centred at the origin .
  6. Using part (ii), or otherwise, show that  .  (2 marks)

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i.     
♦ Mean mark (i) 50%.
 
 
 

 

ii.   

♦ Mean mark (ii) 42%.

 

 

 

iii. 

♦♦♦ Mean mark (iii) 14%.