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

Two forces act at a point in the xy-plane: F₁ = 130 N at 10° and F₂ = 210 N at 295°, 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 = 130 N, θ₁ = 10°F₂ = 210 N, θ₂ = 295°

φ = °

2.Easy

A force of 300 N acts in the xy-plane at 20° measured anticlockwise from the positive x-axis. Find its y-component F_y.

F = 300 Nθ = 20°

F_y = N

3.Medium

Find the scalar projection comp_b a of a = (−2, −2, −2) onto b = (4, −4, −2), that is, the component of a in the direction of b.

a = (−2, −2, −2)b = (4, −4, −2)

comp_b a =

4.Medium

A constant force F = (10, −30, −5) N acts on an object while it moves in a straight line from point A(−2, −5, −2) to point B(−5, −5, 4). Coordinates are in metres. Find the work W done by the force.

F = (10, −30, −5) NA(−2, −5, −2) mB(−5, −5, 4) m

W = J

5.Hard

The unit vector n̂ is perpendicular to both a = (−4, −1, 4) and b = (−4, 5, −2) and has a positive z-component. Find its y-component n̂_y.

a = (−4, −1, 4)b = (−4, 5, −2)n̂_z > 0

n̂_y =

6.Hard

For a = (−2, 3, 1) and b = (−1, −5, 0), find the z-component (a × b)_z of the cross product a × b.

a = (−2, 3, 1)b = (−1, −5, 0)

(a × b)_z =

Answer key

1. φ = 322.3 °
2. F_y = 102.6 N
3. comp_b a = 0.6667
4. W = −60 J
5. n̂_y = 0.6247
6. (a × b)_z = 13

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