Worksheet: Cross Product in 3D

In this worksheet, we will practice finding the cross product of two vectors in space and how to use it to find the area of geometric shapes.

Q1:

Let Vi= and Wijk=3+2+4. Calculate VW×.

  • A 0 , 4 , 2
  • B 2 , 0 , 0
  • C 3 , 0 , 0
  • D 2 , 3 , 0
  • E 4 , 0 , 3

Q2:

Given that Aijk=9+3 and Bijk=327, determine AB×.

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

Q3:

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

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

Q4:

Given that Aijk=3+35 and Bijk=3+5, determine (4)×(2)AB.

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

Q5:

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

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

Q6:

V and W are two vectors, where V=2,1,4 and W=1,2,0. Calculate VW×.

  • A 4 , 2 , 1 2
  • B 4 , 2 , 9
  • C 8 , 4 , 3
  • D 2 , 2 , 0
  • E 8 , 4 , 5

Q7:

V and W are two vectors, where Vijk=+2+ and Wijk=3+6+3. Calculate VW×.

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

Q8:

V and W are two vectors, where V=7,2,10 and W=2,6,4. Calculate VW×.

  • A 1 4 , 1 2 , 4 0
  • B 3 4 , 5 4 , 6 4
  • C 5 2 , 8 , 3 8
  • D 2 8 , 4 , 4 0
  • E 6 8 , 4 8 , 3 8

Q9:

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

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

Q10:

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

  • A 1 3 , 2 3 , 2 3 or 13,23,23
  • B 2 4 , 4 8 , 4 8 or 24,48,48
  • C 1 , 2 , 2 or 1,2,2
  • D 8 , 1 6 , 1 6 or 8,16,16

Q11:

V and W are two vectors, where V=5,1,2 and W=4,4,3. Calculate VW×.

  • A 5 , 2 3 , 2 4
  • B 1 1 , 2 3 , 1 6
  • C 2 0 , 4 , 6
  • D 2 3 , 2 6 , 4
  • E 3 5 , 1 , 1 6

Q12:

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

Q13:

Let V=1,3,2 and W=7,2,10. Calculate VW×.

  • A 7 , 6 , 2 0
  • B 3 4 , 2 4 , 1 9
  • C 1 2 , 3 6 , 1 0
  • D 2 6 , 4 , 1 9
  • E 3 2 , 2 7 , 1 7

Q14:

Assuming that (,,)ijk form a right-hand system, Aij=16+4, Bij=19+8, and A and B form two adjacent sides of a triangle, find the vector product of A into B and the area of the triangle drawn.

  • A A B k × = 2 0 4 , area =102 square units
  • B A B k × = 5 2 , area =26 square units
  • C A B k × = 3 3 6 , area =168 square units
  • D A B k × = 8 8 , area =44 square units
  • E A B k × = 2 7 2 , area =136 square units

Q15:

Given that the area of 𝐴𝐵𝐶 is 340, what is ||𝐵𝐴×𝐵𝐶||?

Q16:

If 𝐴𝐵𝐶 is a triangle of area 248.5 cm2, find the value of ||𝐵𝐴×𝐴𝐶||.

Q17:

Given that 𝐷=(0,2,8), 𝐸=(6,4,6), and 𝐹=(4,9,2), determine the area of the triangle 𝐷𝐸𝐹 approximated to the nearest hundredth.

Q18:

Triangle 𝐴𝐵𝐶 has vertices 𝐴(5,4), 𝐵(1,5), and 𝐶(3,2). Use vectors to determine its area.

Q19:

Suppose that A=1,1,3 and B=4,8,8 fix two sides of a triangle. What is the area of this triangle, to the nearest hundredth?

Q20:

If A=5,0,1 and B=3,1,3, find AAB×(2).

  • A 1 1 + 2 + 7 i j k
  • B 1 5 5 i j k
  • C 2 + 2 4 + 1 0 i j k
  • D i j k + 1 2 + 5

Q21:

If Ajk=10+5 and Bijk=4+9+, find |5×|BA.

  • A 2 5 2 0 1
  • B 2 5 1 2 9
  • C 1 0 1 2 9
  • D 1 0 2 0 1

Q22:

If A=2,4,1, B=4,1,5, and C=0,3,0, find ABC×(+).

  • A 1 6 1 4 + 2 4 i j k
  • B 1 6 + 1 4 2 4 i j k
  • C 8 1 6 5 i j k
  • D 8 + 1 6 5 i j k

Q23:

Given that A=9,5,1, B=7,𝑘,5, C=10,55,𝑚3, and 𝐴𝐵C, find 𝑘𝑚.

Q24:

Given that the vectors (6,,1)k and (12,6,2) are parallel, determine the value of k.

Q25:

If A and B are unit vectors and 𝜃 the measure of the angle between them, find |()×(+)|ABAB.

  • A s i n 𝜃
  • B 2 𝜃 s i n
  • C 𝐴 𝐵 𝜃 s i n
  • D 2 𝐴 𝐵 𝜃 s i n
  • E 𝐴 𝐵 𝜃 s i n

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