A vector is something that has both a magnitude and a direction. In the text a vector is represented by a bold letter, such as A or B. Many quantities, such as velocity, acceleration, and force, are vectors, and you will need to know how to work with them mathematically.

Components of vectors

We will often split vectors up into components. This is done using the geometry of the right-angled triangle.

The dot product

Taking the dot product of two vectors results in a scalar. Dot products are important in a case such as work, when the work done by a force in moving an object depends on the component of the force in the direction of the displacement.

W = F • d = |F| |d| cos(θ)

Note that the dot product is zero when the vectors are perpendicular to one another, and maximum when they are parallel.

The cross product

Taking the cross product results in a vector that is perpendicular to the two vectors in the cross product. An example is a torque:

τ = r × f
The magnitude of the resultant vector is |r| |F| sin(θ).
The direction is given by the right-hand rule. Using your right hand, point your fingers in the direction of the first vector (r), curl them into the direction of the second vector (F), and your thumb, sticking out, will point in the direction of the resultant vector.