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 = 60 N at 125° and F₂ = 140 N at 85°, 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 = 60 N, θ₁ = 125°F₂ = 140 N, θ₂ = 85°

φ = °

2.Easy

Two forces act at a point in the xy-plane: F₁ = 150 N at 90° and F₂ = 90 N at 0°, both measured anticlockwise from the positive x-axis. Find the magnitude R of their resultant.

F₁ = 150 N, θ₁ = 90°F₂ = 90 N, θ₂ = 0°

R = N

3.Medium

Find the scalar projection comp_b a of a = (7, 4, −3) onto b = (3, −2, 6), that is, the component of a in the direction of b.

a = (7, 4, −3)b = (3, −2, 6)

comp_b a =

4.Medium

Find the scalar projection comp_b a of a = (−3, 5, −7) onto b = (6, −2, 4), that is, the component of a in the direction of b.

a = (−3, 5, −7)b = (6, −2, 4)

comp_b a =

5.Hard

Find the volume V of the parallelepiped whose edges from one corner are a = (−3, 0, −5), b = (2, 4, −5) and c = (−3, 4, 5).

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

V =

6.Hard

For a = (−1, −2, 0) and b = (6, −1, 0), find the z-component (a × b)_z of the cross product a × b.

a = (−1, −2, 0)b = (6, −1, 0)

(a × b)_z =

Answer key

1. φ = 96.72 °
2. R = 174.9 N
3. comp_b a = −0.7143
4. comp_b a = −7.483
5. V = 220
6. (a × b)_z = 13

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