Worksheet: Cross Product in 2D

In this worksheet, we will practice finding the cross product of two vectors in the coordinate plane.

Q1:

All the sides of rhombus 𝑂 𝐡 𝐢 𝐴 are of length 5. Suppose that s i n ∠ 𝐴 𝑂 𝐡 = 3 4 and that 𝐴 𝐡 > 𝑂 𝐢 . Use vector multiplication to find the lengths of the two diagonals.

  • A 𝑂 𝐢 = 4 . 1 1 , 𝐴 𝐡 = 9 . 1 1
  • B 𝑂 𝐢 = 1 6 . 9 3 , 𝐴 𝐡 = 2 6 . 5 4
  • C 𝑂 𝐢 = 1 . 8 4 , 𝐴 𝐡 = 4 . 0 8
  • D 𝑂 𝐢 = 4 . 1 1 , 𝐴 𝐡 = 3 . 2 7

Q2:

Rectangle 𝐴 𝐡 𝐢 𝐷 has vertices 𝐴 ( βˆ’ 6 , βˆ’ 7 ) , 𝐡 ( 0 , 2 ) , 𝐢 ( 6 , βˆ’ 2 ) , and 𝐷 ( 0 , βˆ’ 1 1 ) . Use vectors to determine its area.

Q3:

Given that A = ⟨ βˆ’ 2 , 7 ⟩ and B = ⟨ 3 , βˆ’ 8 ⟩ , determine the area of the parallelogram whose adjacent sides are represented by A and B .

Q4:

If | Γ— | + | β‹… | = 1 7 , 4 2 4 A B A B   and | | = 1 2 A , find | | B .

Q5:

𝐴 𝐡 𝐢 𝐷 is a square of side 4, and u is a unit vector perpendicular to the square’s plane. Find οƒ  𝐴 𝐷 Γ— οƒŸ 𝐡 𝐢 .

Q6:

In the rectangle 𝐴 𝐡 𝐢 𝐷 shown in the figure, calculate οƒ  𝐷 𝐴 Γ— οƒ  𝐡 𝑀 if { , , } i j k form a right-hand system of unit vectors.

  • A βˆ’ 6 4 k
  • B 3 2 k
  • C 6 4 k
  • D βˆ’ 3 2 k

Q7:

If A = ⟨ βˆ’ 7 , 3 ⟩ , B = ⟨ βˆ’ 7 , 7 ⟩ , A C βŠ™ = βˆ’ 7 5 , and C B k Γ— = 3 5 , find C .

  • A ⟨ 9 , βˆ’ 1 1 ⟩
  • B ⟨ 1 , βˆ’ 4 ⟩
  • C ⟨ βˆ’ 1 , βˆ’ 4 ⟩
  • D ⟨ 9 , βˆ’ 4 ⟩

Q8:

Given that A i j = 7 + 2 , B i j = βˆ’ + 2 , and C i j = 6 + 6 , determine ( + ) Γ— ⃑ 𝐡 C A .

  • A 3 0 k
  • B 3 4 k
  • C βˆ’ 3 0 k
  • D 3 k

Q9:

If A i j = βˆ’ βˆ’ 2 , B i j = βˆ’ 4 βˆ’ 4 , A C k Γ— = βˆ’ 3 , and C B k Γ— = 4 , find C .

  • A βˆ’ 5 βˆ’ i j
  • B i j + 5
  • C βˆ’ 2 βˆ’ i j
  • D i j βˆ’

Q10:

If A i j = 3 βˆ’ 5 , B i j = π‘š + 5 , and A B k Γ— = 5 0 , find the value of π‘š .

Q11:

Given that 𝐴 𝐡 𝐢 𝐷 is a square with side length 27 cm, and e is the unit vector perpendicular to its plane, determine οƒ  𝐴 𝐡 Γ— οƒ  𝐢 𝐴 .

Q12:

Given that 𝐴 𝐡 𝐢 𝐷 is a square with side 49, and e is the unit vector perpendicular to its plane, determine οƒ  𝐡 𝐷 Γ— οƒ  𝐴 𝐢 .

Q13:

If 𝐴 𝐡 𝐢 𝐷 is a square with a side length of 81 cm, and e is a unit vector perpendicular to its plane, find οƒ  𝐴 𝐡 Γ— οƒŸ 𝐡 𝐢 .

  • A 6 , 5 6 1 e
  • B 1 6 2 e
  • C 1 3 , 1 2 2 e
  • D 8 1 e

Q14:

𝐴 𝐡 𝐢 𝐷 is a rectangle where C is a unit vector perpendicular to its plane. Find οƒ  𝐴 𝐡 Γ— οƒ  𝐡 𝐷 .

  • A 7 8 4 C
  • B βˆ’ 7 8 4 C
  • C βˆ’ 1 , 2 6 0 C
  • D 1 , 2 6 0 C

Q15:

𝐴 𝐡 𝐢 𝐷 is a rectangle where C is a unit vector perpendicular to its plane. Find οƒ  𝐢 𝑀 Γ— οƒŸ 𝐢 𝐡 .

  • A βˆ’ 6 0 5 √ 2 C
  • B 7 2 6 C
  • C 6 0 5 √ 2 C
  • D βˆ’ 7 2 6 C

Q16:

Given that A = ⟨ π‘˜ , βˆ’ 2 ⟩ , and B = ⟨ βˆ’ 2 0 , βˆ’ 1 0 ⟩ , and A B βˆ₯ , find the value of π‘˜ .

  • A π‘˜ = 1
  • B π‘˜ = 4
  • C π‘˜ = 1 0 0
  • D π‘˜ = βˆ’ 1 0 0
  • E π‘˜ = βˆ’ 4

Q17:

Find the value of | Γ— | + | β‹… | 2 | | | | A B A B A B     .

  • A 1 4
  • B1
  • C 1 2
  • D0
  • E2

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