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 125° and F₂ = 260 N at 75°, 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 = 125°F₂ = 260 N, θ₂ = 75°

φ = °

2.Easy

Find the magnitude |a| of the vector a = (3, −4, −6).

a = (3, −4, −6)

|a| =

3.Medium

A constant force F = (5, −25, −5) N acts on an object while it moves in a straight line from point A(0, 5, 5) to point B(1, −3, 3). Coordinates are in metres. Find the work W done by the force.

F = (5, −25, −5) NA(0, 5, 5) mB(1, −3, 3) m

W = J

4.Medium

Find the angle θ between the vectors a = (−6, 1, 2) and b = (−6, 3, 2), in degrees.

a = (−6, 1, 2)b = (−6, 3, 2)

θ = °

5.Hard

A force F = (−60, −10, 0) N acts at point A(−3, −4, −1). Coordinates are in metres. Find the z-component M_z of the moment of the force about point O(−2, 2, 3).

F = (−60, −10, 0) NA(−3, −4, −1) mO(−2, 2, 3) m

M_z = N·m

6.Hard

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

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

n̂_x =

Answer key

1. φ = 91.16 °
2. |a| = 7.81
3. W = 215 J
4. θ = 16.39 °
5. M_z = −350 N·m
6. n̂_x = 0.5235

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