Worksheet: Products of Vectors

In this worksheet, we will practice calculating the vector product of two vectors and using the vector product to find the angle between two vectors.

Q1:

Calculate the product ⃑ 𝐢 Γ— ⃑ 𝐹 .

  • A 1 1 5 ⃑ π‘˜
  • B 1 4 4 ⃑ π‘˜
  • C 1 8 0 ⃑ π‘˜
  • D 1 5 8 ⃑ π‘˜
  • E 1 0 8 ⃑ π‘˜

Q2:

For the vectors shown in the accompanying diagram, the positive @ π‘₯ -axis corresponds to horizontally right and the positive @ 𝑦 -axis corresponds to vertically upwards.

Find the component of vector οƒ  @ 𝐴 along vector οƒ  @ 𝐢 .

  • A7.13
  • B6.46
  • C8.49
  • D8.66
  • E9.12

Find the component of vector οƒ  @ 𝐢 along vector οƒ  @ 𝐴 .

  • A10.4
  • B11.0
  • C9.35
  • D8.73
  • E10.1

Find the component of vector οƒŸ @ 𝑖 along vector οƒ  @ 𝐹 .

  • A0.909
  • B0.708
  • C0.866
  • D0.783
  • E0.810

Find the component of vector οƒ  @ 𝐹 along vector οƒŸ @ 𝑖 .

  • A17.3
  • B18.0
  • C16.8
  • D13.8
  • E15.1

Q3:

Consider the vectors ⃑ 𝐴 = βˆ’ ⃑ 𝑖 βˆ’ ⃑ 𝑗 and ⃑ 𝐡 = βˆ’ 3 ⃑ 𝑖 βˆ’ ⃑ 𝑗 .

Find ⃑ 𝐴 Γ— ⃑ 𝐡 .

  • A βˆ’ 3 ⃑ π‘˜
  • B ⃑ π‘˜
  • C 2 ⃑ π‘˜
  • D βˆ’ 2 ⃑ π‘˜
  • E 3 ⃑ π‘˜

Find | ⃑ 𝐴 Γ— ⃑ 𝐡 | .

Find the angle between ⃑ 𝐴 and ⃑ 𝐡 .

Find the angle between ( ⃑ 𝐴 Γ— ⃑ 𝐡 ) and ⃑ 𝐢 = ⃑ 𝑖 + ⃑ π‘˜ .

Q4:

A convoy of vehicles has a velocity vector ⃑ 𝑣 = ο€Ί 4 . 0 ⃑ 𝑖 + 3 . 0 ⃑ 𝑗  / k m h .

What is the unit vector of the convoy’s direction of motion?

  • A 0 . 6 ⃑ 𝑖 + 0 . 9 ⃑ 𝑗
  • B 4 . 0 ⃑ 𝑖 + 3 . 0 ⃑ 𝑗
  • C 1 . 8 ⃑ 𝑖 + 2 . 3 ⃑ 𝑗
  • D 0 . 8 ⃑ 𝑖 + 0 . 6 ⃑ 𝑗
  • E 2 . 0 ⃑ 𝑖 + 1 . 5 ⃑ 𝑗

At what angle north of east does the convoy move?

Q5:

The positive π‘₯ -axis is horizontal to the right for the vectors shown.

What is the vector product A C Γ— ?

  • A 1 2 0 k
  • B βˆ’ 1 2 5 k
  • C βˆ’ 1 1 7 k
  • D βˆ’ 1 2 0 k
  • E βˆ’ 1 1 6 k

What is the vector product A F Γ— ?

  • A βˆ’ 1 7 3 k
  • B 1 7 7 k
  • C βˆ’ 1 8 0 k
  • D βˆ’ 1 8 5 k
  • E βˆ’ 1 7 5 k

What is the vector product D C Γ— ?

  • A βˆ’ 1 7 5 k
  • B βˆ’ 1 8 5 k
  • C 9 3 . 7 k
  • D βˆ’ 1 8 0 k
  • E 1 7 7 k

What is the vector product A F C Γ— ( + 2 ) ?

  • A βˆ’ 4 1 3 k
  • B 4 1 6 k
  • C βˆ’ 4 2 1 k
  • D 4 0 3 k
  • E βˆ’ 4 0 8 k

What is the vector product i B Γ— ?

  • A βˆ’ 3 9 k
  • B 3 7 k
  • C 4 4 k
  • D 4 0 k
  • E βˆ’ 4 2 k

What is the vector product j B Γ— ?

  • A βˆ’ 1 0 0 k
  • B 1 1 0 k
  • C βˆ’ 9 0 k
  • D βˆ’ 1 2 0 k
  • E 9 3 k

What is the vector product ( 3 βˆ’ ) Γ— i j B ?

  • A βˆ’ 1 1 0 k
  • B 1 5 0 k
  • C βˆ’ 1 6 0 k
  • D βˆ’ 1 3 0 k
  • E 1 4 0 k

Q6:

Calculate the product ⃑ 𝐴 Γ— ⃑ 𝐡 .

  • A 2 4 . 2 ⃑ π‘˜
  • B 4 4 . 0 ⃑ π‘˜
  • C 2 8 . 8 ⃑ π‘˜
  • D 4 0 . 0 ⃑ π‘˜
  • E 3 4 . 0 ⃑ π‘˜

Q7:

Three dogs pull on a stick, all in different directions, exerting forces ⃑ 𝐹  , ⃑ 𝐹  , and ⃑ 𝐹  . ⃑ 𝐹 = ( 1 0 . 0 ⃑ 𝑖 βˆ’ 2 0 . 4 ⃑ 𝑗 + 2 . 0 ⃑ π‘˜ )  N, ⃑ 𝐹 = ( βˆ’ 1 5 . 0 ⃑ 𝑖 βˆ’ 6 . 2 ⃑ π‘˜ )  N, and ⃑ 𝐹 = ( 5 . 0 ⃑ 𝑖 + 1 2 . 5 ⃑ 𝑗 )  N. The forces ⃑ 𝐹  , ⃑ 𝐹  , and ⃑ 𝐹  apply the displacement vector ⃑ 𝐷 = ( βˆ’ 7 . 9 ⃑ 𝑗 βˆ’ 4 . 2 ⃑ π‘˜ ) cm to the stick.

What is the angle between ⃑ 𝐹  and ⃑ 𝐹  ?

What magnitude of work is done by ⃑ 𝐹  ?

What magnitude of work is done by ⃑ 𝐹  ?

Q8:

Calculate the product F C β‹… .

Q9:

Calculate the product ⃑ 𝐴 β‹… ⃑ 𝐡 .

Q10:

Find the angle between the two vectors ⃑ 𝑦 = ο€» 2 . 0 ⃑ 𝑖 + 4 . 0 ⃑ 𝑗 + 8 . 0 ⃑ π‘˜  and ⃑ 𝑧 = ο€» 6 . 0 ⃑ 𝑖 + 4 . 0 ⃑ 𝑗 + 6 . 0 ⃑ π‘˜  .

Q11:

What is the relationship between the directions of two vectors for which the dot product is zero?

  • AThey are coplanar.
  • BThey are parallel.
  • CThey form a 45 degrees angle with respect to one another.
  • DThey are perpendicular (orthogonal).
  • EThey are antiparallel.

Q12:

Calculate the dot product of y i j k = ( 2 + 4 + 8 ) and z i j k = ( 6 + 4 + 2 ) . Which of the following matches the result?

  • A16
  • B12
  • C72
  • D44
  • E38

Q13:

Calculate the cross product of y i j k = ( 2 + 4 + 8 ) and z i j k = ( 6 + 4 + 2 ) . Which of the following best matches the result?

  • A ( 8 + 4 8 + 8 ) i j k
  • B ( 2 8 + 1 2 + 8 ) i j k
  • C ( βˆ’ 3 2 βˆ’ 4 βˆ’ 2 4 ) i j k
  • D ( βˆ’ 2 4 + 4 4 βˆ’ 1 6 ) i j k
  • E ( 2 βˆ’ 1 2 + 8 ) i j k

Q14:

Which of the following is closest to the angle between the two vectors ⃑ 𝑦 = ( 2 ⃑ 𝑖 + 4 ⃑ 𝑗 + 8 ⃑ π‘˜ ) and ⃑ 𝑧 = ( 6 ⃑ 𝑖 + 4 ⃑ 𝑗 + 2 ⃑ π‘˜ ) ?

  • A 70 degrees
  • B 55 degrees
  • C 32 degrees
  • D 50 degrees
  • E 5 degrees

Q15:

Two displacement vectors A and F have magnitudes of 10.0 m and 20.0 m respectively. The directions of A and F make counterclockwise angles of 3 5 ∘ and 1 1 0 ∘ ,respectively, with the positive π‘₯ -axis, as shown in the accompanying diagram. Find the product A F β‹… . Answer to three significant figures.

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