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

Find the z-component û_z of the unit vector û that points in the direction of a = (−5, 2, −4).

a = (−5, 2, −4)

û_z =

2.Easy

Find the magnitude |a| of the vector a = (6, 6, 8).

a = (6, 6, 8)

|a| =

3.Medium

Find the scalar projection comp_b a of a = (−4, 0, 3) onto b = (−1, −2, −4), that is, the component of a in the direction of b.

a = (−4, 0, 3)b = (−1, −2, −4)

comp_b a =

4.Medium

Find the value of k for which the vectors a = (6, −1, −2) and b = (1, 1, k) are perpendicular.

a = (6, −1, −2)b = (1, 1, k)

k =

5.Hard

Find the area A of the triangle with corners A(−1, 3, 4), B(−3, 1, 2) and C(0, 2, −3). Coordinates are in metres.

A(−1, 3, 4)B(−3, 1, 2)C(0, 2, −3)

A = m²

6.Hard

The unit vector n̂ is perpendicular to both a = (−4, 5, 5) and b = (3, −5, 1) and has a positive z-component. Find its y-component n̂_y.

a = (−4, 5, 5)b = (3, −5, 1)n̂_z > 0

n̂_y =

Answer key

1. û_z = −0.5963
2. |a| = 11.66
3. comp_b a = −1.746
4. k = 2.5
5. A = 10.2 m²
6. n̂_y = 0.5298

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