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.

New worksheetPractise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________   Date: ____________

1.Easy
xyza

Find the x-component ûx of the unit vector û that points in the direction of a = (−5, 7, 0).

a = (−5, 7, 0)
ûx =
2.Easy
xy30°70°F1​F2​

Two forces act at a point in the xy-plane: F₁ = 120 N at 30° and F₂ = 40 N at 70°, 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 = 120 N, θ1 = 30°F2 = 40 N, θ2 = 70°
φ = °
3.Medium
xyzab

Find the value of k for which the vectors a = (5, 1, −1) and b = (0, k, 3) are perpendicular.

a = (5, 1, −1)b = (0, k, 3)
k =
4.Medium
xyzab

Find the dot product a·b of a = (5, 5, −1) and b = (7, −3, 5).

a = (5, 5, −1)b = (7, −3, 5)
a·b =
5.Hard
xyzABC

Find the area A of the triangle with corners A(1, −2, 0), B(−4, −4, −1) and C(3, −2, 1). Coordinates are in metres.

A(1, −2, 0)B(−4, −4, −1)C(3, −2, 1)
A = m²
6.Hard
xyzab

The unit vector n̂ is perpendicular to both a = (−2, 5, 0) and b = (4, −4, 3) and has a positive z-component. Find its x-component n̂x.

a = (−2, 5, 0)b = (4, −4, 3)n̂z > 0
n̂x =

Answer key

  1. 1. ûx = −0.5812
  2. 2. φ = 39.69 °
  3. 3. k = 3
  4. 4. a·b = 15
  5. 5. A = 2.693 m²
  6. 6. n̂x = −0.7454

Full step-by-step solutions: practise this topic on examproblems.com.