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 = 300 N at 90° and F₂ = 260 N at 310°, 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 = 300 N, θ₁ = 90°F₂ = 260 N, θ₂ = 310°

φ = °

2.Easy

Find the y-component û_y of the unit vector û that points in the direction of a = (−5, 2, −1).

a = (−5, 2, −1)

û_y =

3.Medium

Find the scalar projection comp_b a of a = (−7, 1, −2) onto b = (−5, −2, −1), that is, the component of a in the direction of b.

a = (−7, 1, −2)b = (−5, −2, −1)

comp_b a =

4.Medium

Find the scalar projection comp_b a of a = (7, 4, 6) onto b = (−3, 0, −2), that is, the component of a in the direction of b.

a = (7, 4, 6)b = (−3, 0, −2)

comp_b a =

5.Hard

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

A(−3, −1, 4)B(−3, −2, −2)C(−2, −1, −2)

A = m²

6.Hard

Find the volume V of the parallelepiped whose edges from one corner are a = (−1, −4, −5), b = (5, 4, 2) and c = (−1, −4, −2).

a = (−1, −4, −5)b = (5, 4, 2)c = (−1, −4, −2)

V =

Answer key

1. φ = 31.1 °
2. û_y = 0.3651
3. comp_b a = 6.39
4. comp_b a = −9.153
5. A = 4.272 m²
6. V = 48

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