Worksheet: Geometric Applications of Vectors

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


𝐴𝐵𝐶𝐷 is a square, in which the coordinates of the points 𝐴, 𝐵, and 𝐶 are (1,8), (3,10), and (5,8). Use vectors to determine the coordinates of the point 𝐷 and the area of the square.

  • A𝐷(3,6), area =8
  • B𝐷(1,10), area =340
  • C𝐷(9,26), area =16
  • D𝐷(7,10), area =8


Trapezoid 𝐴𝐵𝐶𝐷 has vertices 𝐴(4,14), 𝐵(4,4), 𝐶(12,4), and 𝐷(12,9). Given that 𝐴𝐵𝐷𝐶 and 𝐴𝐵𝐶𝐵, find the area of that trapezoid.


Given a trapezoid 𝐴𝐵𝐶𝐷, in which 𝐴𝐷𝐵𝐶 and 𝐴𝐷𝐵𝐶=7, find the value of 𝑘 such that 𝐴𝐶+𝐵𝐷=𝑘𝐴𝐷.

  • A87
  • B8
  • C17
  • D157


𝐴𝐵𝐶𝐷 is a rectangle, in which the coordinates of the points 𝐴, 𝐵, and 𝐶 are (18,2), (18,3), and (8,𝑘), respectively. Use vectors to find the value of 𝑘 and the coordinates of point 𝐷.

  • A𝑘=1, 𝐷(8,3)
  • B𝑘=2, 𝐷(28,2)
  • C𝑘=3, 𝐷(8,2)
  • D𝑘=1, 𝐷(28,2)
  • E𝑘=2, 𝐷(8,2)


Given the information in the diagram below, find the value of 𝑛 such that 𝐴𝐷+𝐷𝐸=𝑛𝐴𝐶.

  • A67
  • B12
  • C12
  • D67


Given a triangle 𝐴𝐵𝐶, in which 𝐴𝐵=7cm, 𝐵𝐶=56cm, and 𝑚𝐴𝐵𝐶=120, use vectors to determine the length of 𝐴𝐶.

  • A2798 cm
  • B757 cm
  • C117 cm
  • D773 cm


Given that 𝐴, 𝐵, 𝐶, and 𝐷 are four collinear points, where 𝐴𝐵𝐵𝐶𝐶𝐷=383, determine the value of 𝑥 which satisfies 𝐵𝐷=𝑥𝐴𝐵.

  • A113
  • B1
  • C83


In the triangle 𝐴𝐵𝐶, 𝐷𝐵𝐶, where 𝐵𝐷𝐷𝐶=23. Given that 3𝐴𝐵+2𝐴𝐶=𝑘𝐴𝐷, find the value of 𝑘.


If 𝐴(5,1), 𝐵(2,5), 𝐶(2,𝑘), and 𝐷(5,4) are vertices of the trapezoid 𝐴𝐵𝐶𝐷, find the value of 𝑘 using vectors.


If 𝐴(9,8), 𝐵(4,2), and 𝐶(1,3) are vertices of the triangle 𝐴𝐵𝐶, find the coordinates of the point of intersection of its medians using vectors.

  • A(8,2)
  • B83,143
  • C(4,1)
  • D43,73
  • E143,3

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