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Vectors, SPEC1 2024 VCAA 4

Consider the vectors    and  ,  where  .

  1. Find the angle between and .   (2 marks)

  2. Find all possible values of such that the dot product of and is equal to the magnitude of the cross product of and .   (2 marks)

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Vectors, SPEC2 2023 VCAA 5

The points with coordinates and all lie in a plane .

  1. Find the vectors and , and hence show that the area of triangle is 1.5 square units.  (2 marks)

  2. Find the shortest distance from point to the line segment .  (2 marks)

A second plane, , has the Cartesian equation  .

  1. At what acute angle does the line given by , intersect the plane ? Give your answer in degrees correct to the nearest degree.  (2 marks)

A line passes through the origin and is normal to the plane . The line intersects at a point .

  1. Write down an equation of the line in parametric form.  (1 mark)

  2. Find the shortest distance from the origin to the plane .  (2 marks)

  3. Find the coordinates of point .  (2 marks)

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b.   

c.     

d.   

e.   

f.   

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b.   


 

♦♦ Mean mark (b) 35%.

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d.   


 

♦ Mean mark (d) 52%.

e.    .


 

f.   

♦ Mean mark (f) 41%.

Vectors, SPEC1 2014 VCAA 1

Consider the vector  , where    and    are unit vectors in the positive directions of the and axes respectively.

  1. Find the unit vector in the direction of  .  (1 mark)

  2. Find the acute angle that    makes with the positive direction of the -axis.  (2 marks)

  3. The vector  .
  4. Given that    is perpendicular to  find the value of .  (2 marks)

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Mean mark part (b) 51%.

b.   
 
   
 

 

c.