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 CFΓ—.

  • A 1 8 0 k
  • B 1 5 8 k
  • C 1 4 4 k
  • D 1 0 8 k
  • E 1 1 5 k

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 upward.

Find the component of vector A along vector i.

Find the component of vector C along vector j.

Find the component of vector i along vector F.

Find the component of vector F along vector i.

Q3:

Consider the vectors Aij=βˆ’βˆ’ and Bij=βˆ’3βˆ’.

Find ABΓ—.

  • A k
  • B 3 k
  • C βˆ’ 3 k
  • D 2 k
  • E βˆ’ 2 k

Find |Γ—|AB.

Find the angle between A and B.

Find the angle between (Γ—)AB and Cik=+.

Q4:

A convoy of vehicles has a velocity vector vij=(4.0+3.0)/kmh.

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

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

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

What is the vector product AFΓ—?

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

What is the vector product DCΓ—?

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

What is the vector product AFCΓ—(+2)?

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

What is the vector product iBΓ—?

  • A 3 . 0 k
  • B βˆ’ 4 . 0 k
  • C 5 . 2 k
  • D βˆ’ 3 . 0 k
  • E 4 . 0 k

What is the vector product jBΓ—?

  • A 5 . 2 k
  • B 3 . 0 k
  • C βˆ’ 4 . 0 k
  • D βˆ’ 3 . 0 k
  • E 4 . 0 k

What is the vector product (3βˆ’)Γ—ijB?

  • A βˆ’ 9 . 0 k
  • B 1 5 k
  • C βˆ’ 1 5 k
  • D 9 . 0 k
  • E 1 6 k

Q6:

Calculate the product ABΓ—.

  • A 4 4 . 0 k
  • B 2 4 . 2 k
  • C 3 4 . 0 k
  • D 2 8 . 8 k
  • E 4 0 . 0 k

Q7:

Three dogs pull on a stick, all in different directions, exerting forces F, F, and F. Fijk=(10.0βˆ’20.4+2.0) N, Fik=(βˆ’15.0βˆ’6.2) N, and Fij=(5.0+12.5) N. The forces F, F, and F apply the displacement vector Djk=(βˆ’7.9βˆ’4.2) 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 οƒ @𝐹⋅@𝐢.

Q9:

Calculate the product ABβ‹….

Q10:

Find the angle between the two vectors yijk=(2.0+4.0+8.0) and zijk=(6.0+4.0+6.0).

Q11:

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

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

Q12:

Calculate the dot product of yijk=(2+4+8) and zijk=(6+4+2). Which of the following matches the result?

  • A44
  • B16
  • C38
  • D12
  • E72

Q13:

Calculate the cross product of yijk=(2+4+8) and zijk=(6+4+2). Which of the following best matches the result?

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

Q14:

Which of the following is closest to the angle between the two vectors yijk=(2+4+8) and zijk=(6+4+2)?

  • A70 degrees
  • B32 degrees
  • C55 degrees
  • D5 degrees
  • E50 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 35∘ and 110∘,respectively, with the positive π‘₯-axis, as shown in the accompanying diagram. Find the product AFβ‹…. Answer to three significant figures.

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