Vectors

Note: Parts of the explanations in this lesson were composed with ideas from Joseph C. Kolecki’s text “An Introduction to Tensors for Students of Physics and Engineering”.

When we specify numbers, both in physics and in life, different types of quantities are necessary. Consider the following types of questions:

  1. How many cookies can I eat?
  2. How far away is the grocery store, so I can buy cookies?
  3. How do I go from here to that grocery store?

We might answer the questions as follows:

  1. Three cookies
  2. Three miles
  3. One mile north and two miles east

The first answer, three cookies, is a “bare number“ — that is, it needs neither units nor direction. It does not need units because it is a counted quantity rather than a measured one and there is no other system of units in which “three cookies” could be represented differently.

The second answer, three miles, is a scalar — that is, it needs to include a specification of units. The distance to the grocery store could equivalently be specified as 4.83 kilometers or in any other units of length, so it is necessary to include information about what measurement system is used.

The third answer, one mile north and two miles east is a vector in component form. It includes both information about measurement systems and what direction to go the distance specified. The component form means that the vector has pieces in two directions: north and east.

Many quantities in physics are vectors. Such quantities include displacement, velocity, acceleration, jerk, momentum, force, and electric field. Sometimes current and area are also treated as vectors. There is also the subtlety of the pseudovector (the result of a cross product operation, which will be explained shortly), which occurs in physics as the angular analogues of linear mechanical quantities.

Vectors, written in bold print to distinguish them from scalars, can be added (a+b), subtracted (a-b), and multiplied in three ways: by a scalar (ka), by another vector via the dot product (a dot b), and by another vector via the cross product (a cross b). There also exists the dyadic product (ab), but that is beyond the scope of this course. An important property to remember is that vectors can be moved about arbitrarily as long as neither their magnitude nor direction changes.

diagram of the right-hand ruleIf the angle between two vectors a and b is theta, then a dot b = ||a||cos(theta)||b||, a scalar. For the same vectors and angle, ||a cross b|| = ||a||sin(theta)||b||, in a direction determined by the right-hand rule: see the illustration to the right for an example with c=a cross b. For those interested, the webcomic xkcd has some alternative suggestions.

The magnitude of a vector, denoted ||a||, is useful in some operations. If the vector is in component form (i.e., a=a_x*i-hat+a_y*j-hat+a_z*k-hat for a three-dimensional vector), ||a|| can be found by ||a||=sqrt(a_x^2+a_y^2+a_z^2).

In some equations, particularly those used in studying fields, a final type of vector is important: the unit vector. Already you have seen the unit vectors i-hat, j-hat, and k-hat representing the unit vectors in the directions of the positive x-, y-, and z-axes. In general, the unit vector for a vector a (denoted a-hat and pronounced “a-hat”) can be found by a-hat=a/||a||. The critical property of a unit vector is that its magnitude is equal to one, so it can be used in an equation to assign a direction to the result without affecting the numeric value.

In diagrams, vectors are typically indicated with an arrow pointing in the vector’s direction. When studying rotation and fields, one frequently works in three dimensions and thus needs to notate vectors pointing in and out of the diagram. Vectors pointing into the diagram are denoted circled x and vectors pointing out of the diagram are denoted circled dot.

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