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 = 280 N at 45° and F₂ = 40 N at 135°, 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 = 280 N, θ₁ = 45°F₂ = 40 N, θ₂ = 135°

φ = °

2.Easy

Find the magnitude |a| of the vector a = (−10, −3).

a = (−10, −3)

|a| =

3.Medium

Find the angle θ between the vectors a = (1, −2, 6) and b = (0, −2, −2), in degrees.

a = (1, −2, 6)b = (0, −2, −2)

θ = °

4.Medium

Find the angle θ between the vectors a = (−3, −2, 0) and b = (4, −5, −6), in degrees.

a = (−3, −2, 0)b = (4, −5, −6)

θ = °

5.Hard

The unit vector n̂ is perpendicular to both a = (−1, −1, 2) and b = (4, 0, 5) and has a positive z-component. Find its x-component n̂_x.

a = (−1, −1, 2)b = (4, 0, 5)n̂_z > 0

n̂_x =

6.Hard

Find the volume V of the parallelepiped whose edges from one corner are a = (−1, 4, 4), b = (3, 1, 4) and c = (−4, −1, 4).

a = (−1, 4, 4)b = (3, 1, 4)c = (−4, −1, 4)

V =

Answer key

1. φ = 53.13 °
2. |a| = 10.44
3. θ = 116.2 °
4. θ = 93.62 °
5. n̂_x = −0.345
6. V = 116

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