Vectors: dot & cross product worksheet with answers

Free printable Vectors: dot & cross product worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
xy75°275°F1​F2​

Two forces act at a point in the xy-plane: F₁ = 300 N at 75° and F₂ = 160 N at 275°, both measured anticlockwise from the positive x-axis. Find the direction φ of their resultant, measured anticlockwise from the positive x-axis (between 0° and 360°).

F1 = 300 N, θ1 = 75°F2 = 160 N, θ2 = 275°
φ = °
2.Easy
xyza

Find the z-component ûz of the unit vector û that points in the direction of a = (4, 4, 7).

a = (4, 4, 7)
ûz =
3.Medium
xyzab

Find the dot product a·b of a = (5, 0, −7) and b = (−2, −6, −7).

a = (5, 0, −7)b = (−2, −6, −7)
a·b =
4.Medium
xyzab

Find the scalar projection compb a of a = (1, −5, −2) onto b = (−3, 5, −6), that is, the component of a in the direction of b.

a = (1, −5, −2)b = (−3, 5, −6)
compb a =
5.Hard
xyzABC

Find the area A of the triangle with corners A(1, 3, −4), B(2, −1, −1) and C(−1, 1, 2). Coordinates are in metres.

A(1, 3, −4)B(2, −1, −1)C(−1, 1, 2)
A = m²
6.Hard
xyzab

For a = (−2, 1, −1) and b = (2, 1, 2), find the y-component (a × b)y of the cross product a × b.

a = (−2, 1, −1)b = (2, 1, 2)
(a × b)y =

Answer key

  1. 1. φ = 54.91 °
  2. 2. ûz = 0.7778
  3. 3. a·b = 39
  4. 4. compb a = −1.912
  5. 5. A = 11.92 m²
  6. 6. (a × b)y = 2

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