Worksheet: Cross Product

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

Q1:

Let and . Calculate .

  • A
  • B
  • C
  • D
  • E

Q2:

Given that 𝐴 = 9 𝑖 𝑗 + 3 𝑘 and 𝐵 = 3 𝑖 2 𝑗 7 𝑘 , determine 𝐴 × 𝐵 .

  • A 2 1 𝑖 5 4 𝑗 + 1 3 𝑘
  • B 𝑖 7 2 𝑗 + 1 5 𝑘
  • C 2 7 𝑖 2 𝑗 2 1 𝑘
  • D 1 3 𝑖 5 4 𝑗 + 2 1 𝑘

Q3:

Given that 𝐴 = ( 5 , 9 , 1 ) and 𝐵 = ( 2 , 1 , 7 ) , find 𝐴 × 𝐵 .

  • A 2 3 𝑖 + 6 2 𝑗 3 7 𝑘
  • B 6 2 𝑖 + 3 7 𝑗 + 2 3 𝑘
  • C 6 4 𝑖 + 3 3 𝑗 1 3 𝑘
  • D 6 2 𝑖 3 7 𝑗 + 2 3 𝑘

Q4:

Given that 𝐴 = 3 𝑖 + 3 𝑗 5 𝑘 and 𝐵 = 𝑖 3 𝑗 + 5 𝑘 , determine 4 𝐴 × 2 𝐵 .

  • A 2 4 𝑖 7 2 𝑗 2 0 0 𝑘
  • B 8 0 𝑗 + 4 8 𝑘
  • C 1 6 0 𝑗 9 6 𝑘
  • D 1 6 0 𝑗 + 9 6 𝑘

Q5:

If 𝐴 = ( 4 , 2 , 9 ) and 𝐵 = ( 4 , 3 , 4 ) , determine 𝐴 × 𝐵 .

  • A 2 0 𝑖 5 2 𝑗 + 1 9 𝑘
  • B 3 5 𝑖 + 2 0 𝑗 + 4 𝑘
  • C 1 6 𝑖 + 6 𝑗 3 6 𝑘
  • D 1 9 𝑖 5 2 𝑗 + 2 0 𝑘

Q6:

If the force 𝐹 = 𝑥 𝑖 + 2 𝑗 is acting at the point 𝐴 ( 9 , 4 ) , where its moment vector about the point 𝐵 ( 8 , 2 ) is 8 𝑘 , determine the value of 𝑥 .

Q7:

Given that the forces 𝐹 = 𝑖 + 𝑚 𝑗 1 , 𝐹 = 2 𝑖 8 𝑗 2 , and 𝐹 = 𝑛 𝑖 1 2 𝑗 3 are three parallel forces, find the values of 𝑚 and 𝑛 .

  • A 𝑚 = 1 6 , 𝑛 = 3
  • B 𝑚 = 4 , 𝑛 = 1 3
  • C 𝑚 = 1 6 , 𝑛 = 1 3
  • D 𝑚 = 4 , 𝑛 = 3

Q8:

and are two vectors, where and . Calculate .

  • A
  • B
  • C
  • D
  • E

Q9:

𝑉 and 𝑊 are two vectors, where 𝑉 = 𝑖 + 2 𝑗 + 𝑘 and 𝑊 = 3 𝑖 + 6 𝑗 + 3 𝑘 . Calculate 𝑉 × 𝑊 .

  • A ( 3 , 6 , 9 )
  • B ( 3 , 1 2 , 3 )
  • C ( 1 2 , 0 , 1 2 )
  • D ( 0 , 0 , 0 )
  • E ( 0 , 6 , 1 2 )

Q10:

and are two vectors, where and . Calculate .

  • A
  • B
  • C
  • D
  • E

Q11:

If 𝐴 = ( 3 , 4 , 4 ) , 𝐵 = ( 2 , 5 , 4 ) , and 𝐶 = ( 4 , 4 , 2 ) , find 𝐴 𝐵 × 𝐶 𝐴 .

  • A 6 𝑖 + 6 𝑗 + 1 5 𝑘
  • B 2 𝑖 2 𝑗 8 𝑘
  • C 2 𝑖 + 1 6 𝑗 + 1 9 𝑘
  • D 6 𝑖 6 𝑗 1 5 𝑘

Q12:

Find the unit vectors that are perpendicular to both 𝐴 = ( 4 , 2 , 0 ) and 𝐵 = ( 4 , 6 , 4 ) .

  • A ( 8 , 1 6 , 1 6 ) or ( 8 , 1 6 , 1 6 )
  • B 1 3 , 2 3 , 2 3 or 1 3 , 2 3 , 2 3
  • C ( 2 4 , 4 8 , 4 8 ) or ( 2 4 , 4 8 , 4 8 )
  • D ( 1 , 2 , 2 ) or ( 1 , 2 , 2 )

Q13:

and are two vectors, where and . Calculate .

  • A
  • B
  • C
  • D
  • E

Q14:

Given that 𝐴 = ( 3 , 4 , 0 ) , and 𝐵 = ( 1 , 5 , 1 ) , determine the unit vector perpendicular to the plane of the two vectors 𝐴 and 𝐵 .

  • A 4 3 8 6 𝑖 3 3 8 6 𝑗 + 1 9 3 8 6 𝑘
  • B 4 1 4 6 𝑖 3 1 4 6 𝑗 + 1 1 1 4 6 𝑘
  • C 1 1 3 8 6 𝑖 + 4 3 8 6 𝑗 + 3 3 8 6 𝑘
  • D 4 1 4 6 𝑖 + 3 1 4 6 𝑗 + 1 1 1 4 6 𝑘

Q15:

Let and . Calculate .

  • A
  • B
  • C
  • D
  • E

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