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Vectors, SPEC1 2022 VCAA 6b

is a semicircle of radius with equation  . is a point on the semicircle , as shown below.

  1. Express the vectors and in terms of  , , , and , where is a unit vector in the direction of the positive -axis and is a unit vector in the direction of the positive -axis.   (1 mark)

  2. Hence, using the vector scalar (dot) product, determine whether is perpendicular to .   (3 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.   

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e.   

f.   

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♦♦ Mean mark (b) 35%.

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♦ Mean mark (d) 52%.

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♦ Mean mark (f) 41%.

Vectors, SPEC1 2023 VCAA 9

A plane contains the points and , where is a real constant. The plane has a -axis intercept of 2 at the point .

  1. Write down the coordinates of point .   (1 mark)
  1. Show that and are   and  , respectively.   (1 mark)
  1. Hence find the equation of the plane in Cartesian form.  (2 marks)
  1.  Find .   (1 mark)
  1.   and are adjacent sides of a parallelogram. Find the area of this parallelogram.   (1 mark)
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Vectors, SPEC1 2013 VCAA 3

The coordinates of three points are 

  1. Find    (1 mark)

  2. The points and are the vertices of a triangle.
  3. Prove that the triangle has a right angle at   (2 marks)

  4. Find the length of the hypotenuse of the triangle.  (1 mark)

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Vectors, SPEC1 2015 VCAA 1

Consider the rhombus    shown below, where    and  , and is a positive real constant.
 

VCAA 2015 spec 1a
 

  1. Find    (1 mark)

  2. Show that the diagonals of the rhombus    are perpendicular.  (2 marks)

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MARKER’S COMMENT: Vector notation was poor in many answers.