Section 11–4 Vectors

(Directed physical quantity / Three numbers / Transformation properties)

In this section, Feynman discusses definitions of a vector from a perspective of a directed physical quantity, three numbers, and transformation properties.

1. Directed physical quantity:

“All quantities that have a direction, like a step in space, are called vectors… suppose there is another directed physical quantity (Feynman et al., 1963, section 11–4 Vectors).”

A vector is commonly defined as a physical quantity that has a direction. Physicists use vectors to represent physical quantities such as velocity which we can keep track of the direction a body is moving, not just its speed. In his Tips on Physics, Feynman adds that “[a] vector is like a push that has a certain direction, or a speed that has a certain direction, or a movement that has a certain direction and it’s represented on a piece of paper by an arrow in the direction of the thing (Feynman et al., 2006, p. 23).” However, we may criticize the definition of a vector that simply includes a direction and magnitude because an arrow has these two properties, but it is not a vector. As a suggestion, physics teachers should clarify that physical quantities, such as velocity and force, have vector properties but they are not vectors.

In general, a vector can be defined geometrically and algebraically. In Wilczek’s (2015) words, “[g]eometrically, a vector is a quantity that has both magnitude and direction (p. 393).” One may explain that the Latin word vector means carrier and thus, a line with an arrow that represents a vector is, in a sense, “carried” from the origin to another point. Specifically, it is a free vector that is not anchored to a particular point (or independent of location). For example, we can use the same free vector v to represent a displacement from O to A, or from A to B. This is different from bound vectors, such as w and x, that are strictly fixed to the points W and X respectively.

2. Three numbers:

“It means three numbers, but not really only those three numbers, because if we were to use a different coordinate system, the three numbers would be changed to x′, y′, and z′ (Feynman et al., 1963, section 11–4 Vectors).”

Feynman states that a vector means three numbers. He elaborates the need to represent a step in space, say from the origin to a particular point P whose location is (x, y, z), by specifying three numbers. More important, a vector can be represented by three numbers: x, y, and z, but it is not really only those three numbers because the three numbers can be changed to x′, y′, and z′ if we use a different coordinate system. In other words, a vector v can be defined by three components (v_x, v_y, v_z) of the vector in a three-dimensional space. The components of a vector allow us to resolve a single vector quantity into three scalar quantities which are related to our physical world (or experience).

According to Wilczek (2015), “[a]lgebraically, a vector is simply a sequence of numbers (p. 393).” This is an algebraic definition of a vector that does not restrict to only three numbers. Broadly speaking, mathematicians’ definition of a vector is an ordered set of pure numbers. This is a general definition that allows a vector to be represented by more than three numbers. To be precise, vectors are mathematical objects that belong to a vector space. Mathematicians conceptualize a vector space as a mathematical structure which requires the concept of linear combination. In general, a linear combination of vectors that is expressed as av₁ + bv₂ + cv₃ +… is still a vector.

3. Transformation properties:

“…it has the same mathematical transformation properties as a ‘step in space’ (Feynman et al., 1963, section 11–4 Vectors).”

It is inadequate to simply define vectors in terms of a directed physical quantity and three numbers. Interestingly, Feynman explains that we can represent a force by an arrow because it has the same mathematical transformation properties as a “step in space.” Mathematically, we can conceptualize a force by using an equation like F = kr and calculate the three components of the force upon a rotation as a geometric problem. In essence, “[v]ectors are defined by their transformation properties with respect to the rotation group, and classified according to their transformation properties with respect to parity (Sozzi, 2008, p. 16).” The importance of transformation properties (when a coordinate system is changed) is missing in many definitions of vectors.

In Arons’ (1990) words, “[t]he final part of the definition resides in behavior with respect to transformation under rotation of coordinate axes, and this behavior is crucial to the final distinction between Cartesian and pseudo-vectors (p. 93).” This is because a pseudo-vector (or axial vector) transforms like a vector under a proper rotation, but it does not obey the commutative law in a three-dimensional space. As an example, the orientation of a book that is transformed by two successive 90° rotations (horizontal axis followed by vertical axis) will be entirely different, if the order of the two rotations is reversed.

Questions for discussion:

1. How would you define a vector geometrically?

2. How would you define a vector algebraically?

3. What are the transformation properties of a vector?

The moral of the lesson: a vector should be defined in terms of a directed physical quantity, an ordered set of pure numbers, and transformation properties.

References:

1. Arons, A. B. (1990). A Guide to Introductory Physics Teaching. New York: Wiley.

2. 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.

3. 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.

4. Sozzi, M. (2008). Discrete Symmetries and CP Violation: From Experiment to Theory. Oxford: Oxford University Press.

5. Wilczek, F. (2015). A Beautiful Question: Finding Nature’s Deep Design. New York: Penguin Press.