Section 11–5 Vector algebra

(Vector addition / Vector subtraction / Vector differentiation)

In this section, Feynman discusses the addition of vectors, subtraction of vectors, and differentiation of vectors.

1. Vector addition:

“Now we must describe the laws, or rules, for combining vectors in various ways. The first such combination is the addition of two vectors… (Feynman et al., 1963, section 11–5 Vector algebra).”

In general, the laws of vector addition can be illustrated geometrically and algebraically. Geometrically, we can add two vectors (a and b) by joining the “tail” (terminal point) of b to the “head” (initial point) of a to form another vector, a + b. Algebraically, if a has the three components (a_x, a_y, a_z) and b has the three components (b_x, b_y, b_z), the resultant vector can be represented by three new numbers (a_x + b_x, a_y + b_y, a_z + b_z). Moreover, a_x + b_x, a_y + b_y, a_z + b_z is an invariant under rotation because physical phenomena are unchanged in different coordinate systems. Specifically, additions of vector obey the commutative law of addition, a + b = b + a, associative law of addition, a + (b + c) = (a + b) + c, and distributive law of addition, a(b + c) = ab + ac.

Feynman explains algebraically the geometric significance of a + b by adding the components of b to the components of a. He illustrates the commutative property of vector additions by placing the “tail” of b to the “head” of a and joining the “tail” of a to the “head” of b to form the vector c; similarly, we can add a to b and obtain the same result by placing the “tail” of a to the “head” of b. In his Tips on Physics, Feynman mentions that “[w]hen we draw a diagram of something like this, it is often convenient to place the tails of the arrows where the forces are applied, even though in general there’s no meaning to the location of vectors (Feynman et al., 2006, p. 24).” In short, forces are free vectors that are not fixed in space. Thus, we can add any two vectors by using the Triangle Law of vector addition or Parallelogram Law of vectors.

2. Vector subtraction:

“Now let us consider vector subtraction. We may define subtraction in the same way as addition, but instead of adding, we subtract the components (Feynman et al., 1963, section 11–5 Vector algebra).”

Feynman defines vector subtraction in the same way as vector addition. Furthermore, we need to define a negative vector, −b =−1b. Therefore, vector subtraction is simply an addition of a negative vector. Algebraically, we can define vector subtraction by subtracting the components of a vector. Geometrically, Feynman illustrates vector subtraction by using a figure to show that d = a – b = a + (−b) and explain that it is equivalent to the relation a = b + d. Importantly, the result of vector subtraction does not depend on whether we first draw the vector a or b: a − b = −b + a. That is, a subtraction of vector also obeys the commutative law, associative law, and distributive law.

Perhaps Feynman realizes his explanation of vector subtraction was inadequate. In one of his review lectures, he adds that “if the total force on the object is known to be F, we have F' + X = F. And so, X = F – F'. Thus to find X you have to take the difference of two vectors, and you can do that in either of two ways: you can take – F' which is a vector in the opposite direction as F' and add it to F. Otherwise, F – F' is simply the vector drawn from the head of to the head of F. Now, the disadvantage of the second method is that you may have a tendency to draw the arrow as shown in Figure 1-5; although the direction and length of the difference is right, the application of the force is not located at the tail of the arrow—so watch out. In case you’re nervous about it, or there’s any confusion, use the first method (Feynman et al., 2006).” In other words, the vector b – a can be obtained geometrically by moving the vector –b to join its initial point to the terminal point of the vector a, or vice versa.

3. Vector differentiation:

“…we see from this argument that if we differentiate any vector with respect to time we produce a new vector (Feynman et al., 1963, section 11–5 Vector algebra).”

Feynman explains vector differentiation by using velocity as an analogy. Essentially, we need to apply the concept of limit to differentiate a displacement vector. It means that the limit, as Δt goes to 0, and the difference between the displacement vectors at the time t + Δt and the time t, is divided by Δt: v = dr/dt. The velocity vector can be expressed in terms of its components, dx/dt, dy/dt, and dz/dt. It is worth mentioning that we produce a new vector if we differentiate any vector with respect to time. Additionally, Feynman states that velocity is a vector because it is the difference of two vectors. However, physicists may clarify that velocity is not a vector, but it has vector properties.

In his Tips on Physics, Feynman elaborates that “we can do what’s called differentiating the vectors. The derivative of a vector with respect to time is meaningless unless the vector depends on the time, of course. That means we have to imagine some vector that is different all the time: as time goes on, the vector keeps changing, and we want the rate of change. For example, the vector A(t) might be the position, at time t, of an object that’s flying around. At the next moment, t', the object has moved from A(t) to A(t'); we would like to calculate the rate of change of A at time t. The rule is the following: that in the interval Dt = t' - t, the thing has moved from A(t) to A(t') , so the displacement is DA = A(t') - A(t), a difference vector from the old position to the new position (Feynman et al., 2006, p. 29).”

Questions for discussion:

1. How would you illustrate the proof of vector addition?

2. How would you explain the proof of vector subtraction?

3. How do you prove that a vector is differentiable?

The moral of the lesson: the addition of vectors, subtraction of vectors, and differentiation of vectors can be proved algebraically and geometrically.

References:

1. Feynman, R. P., Gottlieb, & M. A., Leighton, R. (2006). Feynman’s tips on physics: reflections, advice, insights, practice: a problem-solving supplement to the Feynman lectures on physics. San Francisco: Pearson Addison-Wesley.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.