Worksheet: Cross Product

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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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