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 𝑂𝐡𝐢𝐴 are of length 5. Suppose that sinβˆ π΄π‘‚π΅=34 and that 𝐴𝐡>𝑂𝐢. Use vector multiplication to find the lengths of the two diagonals.

  • A𝑂𝐢=1.84, 𝐴𝐡=4.08
  • B𝑂𝐢=4.11, 𝐴𝐡=9.11
  • C𝑂𝐢=16.93, 𝐴𝐡=26.54
  • D𝑂𝐢=4.11, 𝐴𝐡=3.27

Q2:

A rectangle 𝐴𝐡𝐢𝐷 has vertices 𝐴(βˆ’6,βˆ’7), 𝐡(0,2), 𝐢(6,βˆ’2), and 𝐷(0,βˆ’11). Use vectors to determine its area.

Q3:

Given that A=βŸ¨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‹…|=17,424ABAB and ||=12A, find ||B.

Q5:

𝐴𝐡𝐢𝐷 is a square of side 4, and u is a unit vector perpendicular to the square’s plane. Find οƒ π΄π·Γ—οƒŸπ΅πΆ.

Q6:

In the rectangle 𝐴𝐡𝐢𝐷 shown in the figure, calculate 𝐷𝐴×𝐡𝑀 if {,,}ijk form a right-hand system of unit vectors.

  • A32k
  • B64k
  • Cβˆ’32k
  • Dβˆ’64k

Q7:

If A=βŸ¨βˆ’7,3⟩, B=βŸ¨βˆ’7,7⟩, ACβŠ™=βˆ’75, and CBkΓ—=35, find C.

  • A⟨9,βˆ’4⟩
  • B⟨1,βˆ’4⟩
  • C⟨9,βˆ’11⟩
  • DβŸ¨βˆ’1,βˆ’4⟩

Q8:

Given that Aij=7+2, Bij=βˆ’+2, and Cij=6+6, determine (+)×⃑𝐡CA.

  • A30k
  • B34k
  • Cβˆ’30k
  • D3k

Q9:

If Aij=βˆ’βˆ’2, Bij=βˆ’4βˆ’4, ACkΓ—=βˆ’3, and CBkΓ—=4, find C.

  • Aij+5
  • Bβˆ’2βˆ’ij
  • Cijβˆ’
  • Dβˆ’5βˆ’ij

Q10:

If Aij=3βˆ’5, Bij=π‘š+5, and ABkΓ—=50, find the value of π‘š.

Q11:

Given that 𝐴𝐡𝐢𝐷 is a square with side length 27 cm, and e is the unit vector perpendicular to its plane, determine 𝐴𝐡×𝐢𝐴.

  • A27e
  • B54e
  • C13e
  • D729e

Q12:

Given that 𝐴𝐡𝐢𝐷 is a square with side 49, and e is the unit vector perpendicular to its plane, determine 𝐡𝐷×𝐴𝐢.

  • Aβˆ’4,802e
  • B4,802e
  • C2,401e
  • Dβˆ’2,401e

Q13:

If 𝐴𝐡𝐢𝐷 is a square with a side length of 81 cm, and e is a unit vector perpendicular to its plane, find οƒ π΄π΅Γ—οƒŸπ΅πΆ.

  • A6,561e
  • B13,122e
  • C162e
  • D81e

Q14:

𝐴𝐡𝐢𝐷 is a rectangle where C is a unit vector perpendicular to its plane. Find 𝐴𝐡×𝐡𝐷.

  • A784C
  • Bβˆ’1,260C
  • Cβˆ’784C
  • D1,260C

Q15:

𝐴𝐡𝐢𝐷 is a rectangle where C is a unit vector perpendicular to its plane. Find οƒ πΆπ‘€Γ—οƒŸπΆπ΅.

  • A605√2C
  • B726C
  • Cβˆ’726C
  • Dβˆ’605√2C

Q16:

Given that A=βŸ¨π‘˜,βˆ’2⟩, and B=βŸ¨βˆ’20,βˆ’10⟩, and ABβˆ₯, find the value of π‘˜.

  • Aπ‘˜=1
  • Bπ‘˜=4
  • Cπ‘˜=βˆ’100
  • Dπ‘˜=βˆ’4
  • Eπ‘˜=100

Q17:

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

  • A14
  • B2
  • C1
  • D12
  • E0

Q18:

Find the area of a triangle 𝐴𝐡𝐢, where 𝐴(βˆ’8,βˆ’9), 𝐡(βˆ’7,βˆ’8), and 𝐢(9,βˆ’2).

Q19:

𝐴𝐡𝐢𝐷 is a rhombus, in which the coordinates of the points 𝐴 and 𝐡 are (5,βˆ’9) and (βˆ’10,12), respectively. Use vectors to determine its perimeter.

  • A3√74 length units
  • B6√74 length units
  • C12√74 length units
  • D666 length units

Q20:

Rhombus 𝐴𝐡𝐢𝐷 has vertices 𝐴(βˆ’4,6), 𝐡(9,2), 𝐢(βˆ’2,10), and 𝐷(βˆ’15,14). Use vectors to determine its area.

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