Worksheet: Geometric Applications of Vectors

In this worksheet, we will practice using vectors operations and vectors properties to solve problems involving geometrical shapes.

Q1:

𝐴 𝐡 𝐢 𝐷 is a square, in which the coordinates of the points 𝐴 , 𝐡 , and 𝐢 are ( 1 , βˆ’ 8 ) , ( 3 , βˆ’ 1 0 ) , and ( 5 , βˆ’ 8 ) . Use vectors to determine the coordinates of the point 𝐷 and the area of the square.

  • A 𝐷 ( 1 , 1 0 ) , area = 3 4 0
  • B 𝐷 ( 9 , βˆ’ 2 6 ) , area = 1 6
  • C 𝐷 ( 7 , βˆ’ 1 0 ) , area = 8
  • D 𝐷 ( 3 , βˆ’ 6 ) , area = 8

Q2:

Trapezium 𝐴 𝐡 𝐢 𝐷 has vertices 𝐴 ( 4 , 1 4 ) , 𝐡 ( 4 , βˆ’ 4 ) , 𝐢 ( βˆ’ 1 2 , βˆ’ 4 ) , and 𝐷 ( βˆ’ 1 2 , 9 ) . Given that οƒ  𝐴 𝐡 βˆ₯ οƒ  𝐷 𝐢 and οƒ  𝐴 𝐡 βŸ‚ οƒŸ 𝐢 𝐡 , find the area of that trapezium.

Q3:

Given a trapezium 𝐴 𝐡 𝐢 𝐷 , in which 𝐴 𝐷 βˆ₯ 𝐡 𝐢 and 𝐴 𝐷 𝐡 𝐢 = 7 , find the value of π‘˜ such that οƒ  𝐴 𝐢 + οƒ  𝐡 𝐷 = π‘˜ οƒ  𝐴 𝐷 .

  • A8
  • B 1 5 7
  • C 1 7
  • D 8 7

Q4:

𝐴 𝐡 𝐢 𝐷 is a rectangle, in which the coordinates of the points 𝐴 , 𝐡 , and 𝐢 are ( βˆ’ 1 8 , βˆ’ 2 ) , ( βˆ’ 1 8 , βˆ’ 3 ) , and ( βˆ’ 8 , π‘˜ ) , respectively. Use vectors to find the value of π‘˜ and the coordinates of point 𝐷 .

  • A π‘˜ = βˆ’ 3 , 𝐷 ( βˆ’ 8 , βˆ’ 2 )
  • B π‘˜ = βˆ’ 1 , 𝐷 ( βˆ’ 2 8 , βˆ’ 2 )
  • C π‘˜ = βˆ’ 2 , 𝐷 ( βˆ’ 8 , βˆ’ 2 )
  • D π‘˜ = βˆ’ 2 , 𝐷 ( βˆ’ 2 8 , βˆ’ 2 )
  • E π‘˜ = βˆ’ 1 , 𝐷 ( βˆ’ 8 , βˆ’ 3 )

Q5:

Given the information in the diagram below, find the value of 𝑛 such that οƒ  𝐴 𝐷 + οƒ  𝐷 𝐸 = 𝑛 οƒ  𝐴 𝐢 .

  • A βˆ’ 6 7
  • B 1 2
  • C 6 7
  • D βˆ’ 1 2

Q6:

Given a triangle 𝐴 𝐡 𝐢 , in which 𝐴 𝐡 = 7 c m , 𝐡 𝐢 = 5 6 c m , and π‘š ∠ 𝐴 𝐡 𝐢 = 1 2 0 ∘ , use vectors to determine the length of 𝐴 𝐢 .

  • A 7 √ 5 7 cm
  • B 7 √ 7 3 cm
  • C 2 √ 7 9 8 cm
  • D 1 1 √ 7 cm

Q7:

Given that 𝐴 , 𝐡 , 𝐢 , and 𝐷 are four collinear points, where 𝐴 𝐡 ∢ 𝐡 𝐢 ∢ 𝐢 𝐷 = 3 ∢ 8 ∢ 3 , determine the value of π‘₯ which satisfies οƒ  𝐡 𝐷 = π‘₯ οƒ  𝐴 𝐡 .

  • A 8 3
  • B 1 1 3
  • C 1

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