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Worksheet: Determinants and Area of Parallelograms

Q1:

Use determinants to calculate the area of the quadrilateral with vertices ( 1 , 1 ) , ( 3 , 2 ) , ( 4 , 1 ) , ( 0 , 3 ) a n d .

  • A31
  • B10
  • C16
  • D 3 1 2
  • E14

Q2:

Use determinants to calculate the area of the polygon with vertices ( 1 , 1 ) , ( 1 , 2 ) , ( 3 , 1 ) , ( 0 , 3 ) , ( 2 , 1 ) a n d .

Q3:

Use determinants to calculate the area of the parallelogram with vertices ( 1 , 1 ) , ( 4 , 5 ) , ( 2 , 8 ) , and ( 3 , 4 ) .

Q4:

Use determinants to calculate the area of the parallelogram with vertices ( 0 , 0 ) , ( 4 , 1 ) , ( 5 , 4 ) , and ( 1 , 3 ) .

Use determinants to calculate the area of the parallelogram with vertices ( 𝑎 , 𝑏 ) , ( 4 + 𝑎 , 1 + 𝑏 ) , ( 5 + 𝑎 , 4 + 𝑏 ) , and ( 1 + 𝑎 , 3 + 𝑏 ) .

Use determinants to calculate the area of the parallelogram with vertices ( 3 , 2 ) , ( 1 , 1 ) , ( 2 , 2 ) , and ( 2 , 1 ) .

Q5:

Use determinants to calculate the area of the polygon with vertices ( 0 , 0 ) , ( 2 , 1 ) , ( 4 , 2 ) , ( 1 , 4 ) , ( 1 , 2 ) a n d .

Q6:

The unit square is defined as the square with vertices ( 0 , 0 ) , ( 1 , 0 ) , ( 1 , 1 ) , and ( 0 , 1 ) . Consider the parallelogram with vertices 𝐴 ( 0 , 0 ) , 𝐵 ( 4 , 2 ) , 𝐶 ( 5 , 5 ) , and 𝐷 ( 1 , 3 ) .

Explain how the parallelogram can be produced from matrix 𝑀 = 4 1 2 3 .

  • A 𝐴 𝐶 𝐵 𝐷 is the image of the unit square after multiplication by 𝑀 .
  • B 𝐷 𝐶 𝐵 𝐴 is the image of the unit square after multiplication by 𝑀 .
  • C 𝐴 𝐵 𝐶 𝐷 is the image of the unit square after multiplication by 𝑀 .

Write the area of 𝐴 𝐵 𝐶 𝐷 as a determinant and determine its value.

  • Aarea ( 𝐴 𝐵 𝐶 𝐷 ) = 𝑀 = 1 0 d e t
  • Barea ( 𝐴 𝐵 𝐶 𝐷 ) = 𝑀 = 1 4 d e t
  • Carea ( 𝐴 𝐵 𝐶 𝐷 ) = 𝑀 = 1 0 d e t
  • Darea ( 𝐴 𝐵 𝐶 𝐷 ) = 𝑀 = 1 4 d e t
  • Earea ( 𝐴 𝐵 𝐶 𝐷 ) = 𝑀 = 2 d e t

Explain how the parallelogram can be produced from matrix 𝑁 = 1 4 3 2 .

  • A 𝐷 𝐶 𝐵 𝐴 is the image of the unit square after multiplication by 𝑁 .
  • B 𝐴 𝐶 𝐵 𝐷 is the image of the unit square after multiplication by 𝑁 .
  • C 𝐴 𝐷 𝐶 𝐵 is the image of the unit square after multiplication by 𝑁 .

Write the area of 𝐴 𝐵 𝐶 𝐷 in terms of the matrix 𝑁 .

  • Aarea ( 𝐴 𝐵 𝐶 𝐷 ) = | 𝑁 | d e t
  • Barea ( 𝐴 𝐵 𝐶 𝐷 ) = 2 | 𝑁 | d e t 2
  • Carea ( 𝐴 𝐵 𝐶 𝐷 ) = 1 2 | 𝑁 | d e t
  • Darea ( 𝐴 𝐵 𝐶 𝐷 ) = 1 4 | 𝑁 | d e t
  • Earea ( 𝐴 𝐵 𝐶 𝐷 ) = 2 | 𝑁 | d e t