Section 11–7 Scalar product of vectors

(Algebraic definition / Geometric definition / Applications of scalar product)

In this section, Feynman discusses an algebraic definition of scalar product, a geometric definition of scalar product, and applications of scalar product.

1. Algebraic definition:

“… we now define the following quantity for any two vectors a and b: a.b = a_xb_x + a_yb_y + a_zb_z (Feynman et al., 1963, section 11–7 Scalar product of vectors).”

Feynman explains that the length of a step in space would be the same in any coordinate system. That is, the distance of a particular step r can be represented by r = √(x²+y²+z²) in a coordinate system and by r′ = √(x′²+y′²+z′²) in another coordinate system such that the distance r = |r| = |r′|. Thus, we can provide an algebraic definition of scalar product (or dot product) for any two vectors a and b: a.b = a_xb_x + a_yb_y + a_zb_z. An important property of scalar product is that the real number produced remains the same in any three mutually perpendicular axes. On the contrary, Feynman changes the order of presentation by first providing a geometric definition of scalar product followed by an algebraic definition in his Tips on Physics (Feynman et al., 2006).

To show that the scalar product a_xb_x+a_yb_y+a_zb_z is invariant in any coordinate systems, one may cite an important fact in which a.a, b.b, and c.c, are also invariant. Feynman suggests a method by expanding the equation: (a_x + b_x)² + (a_y + b_y)² + (a_z + b_z)² = (a_x′ + b_x′)² + (a_y′ + b_y′)² +(a_z′ + b_z′)². As a result, there will be sums of squares of the components of a and b as well as cross product terms such as a_xb_x and a_yb_y. However, it is relatively easy to prove that the scalar product is invariant in all coordinate systems by showing that a_xb_x + a_yb_y + a_zb_z = ½(|a + b|² – |a|² – |b|²). As another alternative, one may prefer to prove |b – a|² = |a|² + |b|² – 2a.b by using the cosine law of triangle (or cosine rule). In a sense, the scalar product of two vectors is tantamount to the cosine law.

2. Geometric definition:

“…geometrical way to calculate a.b, without having to calculate the components of a and b: a.b is the product of the length of a and the length of b times the cosine of the angle between them (Feynman et al., 1963, section 11–7 Scalar product of vectors).”

The scalar product of two vectors, a.b, can be geometrically defined as a product of the length of a and the length of b times the cosine of the angle between them. Feynman elaborates that we can choose a special coordinate system in which the x-axis lies along a such that the only component of a is a_x, which is the whole length of a. Thus, the equation a.b = a_xb_x+a_yb_y+a_zb_z is shortened to a.b = a_xb_x and this is simply the length of a times the component of b in the direction of a, that is, a(bcos θ). One may add that the scalar product of two vectors is closely related to the cosine law of triangle. Mathematically, the cosines of the angles (α, β, γ) between a vector and the three coordinate axes can be expressed as cos² α + cos² β + cos² γ = 1.

Succinctly, Feynman explains that if a.b = abcos θ is true in one coordinate system, it is true in all because a.b is independent of the coordinate system. Physics teachers should realize that the scalar product of two vectors is now expressed in polar coordinates instead of Cartesian coordinates. When the scalar product is expressed in polar coordinates, it is equal to the product of their lengths, multiplied by the cosine of the angle between them. One may also prefer Feynman’s additional explanation in his Tips on Physics: “[i]t is evident that since |A| cos θ is the projection of A onto B, A.B is equal to the projection of A onto B times the magnitude of B. Similarly, since |B| cos θ is the projection of B onto A, A.B also equals the projection of B onto A times the magnitude of A (Feynman et al., 2006, p. 28).”

3. Applications of scalar product:

“We have not yet defined work, but it is equivalent to the energy change, the weights lifted, when a force F acts through a distance s: Work=F.s (Feynman et al., 1963, section 11–7 Scalar product of vectors).”

Curiously, Feynman asks the question “[w]hat good is the dot product?” without first explaining the meaning of dot product. He admits that he has not defined work and states that it is equivalent to the energy change, that is, when a force F acts through a displacement s: Work = F.s. Importantly, the work done by a force is defined as the product of the force and the parallel distance over which it acts. In short, the term work is misleading to students and it may be rephrased as “force-displacement product” or “dot product of force and parallel distance”. This is in contrast to the term torque that is a “cross product of force and perpendicular distance. Note that a torque is dependent on the magnitude of the force applied and its perpendicular distance from the axis of rotation.

Feynman ends the chapter by discussing unit vector whose dot product with itself is equal to unity. The dot products of unit vectors are summarized as i.i = 1, i.j = 0, j.j = 1, i.k = 0, j.k = 0, and k.k = 1. By applying definitions of unit vectors, we can represent physical quantities such as forces by writing their components in the form of a = a_xi + a_yj + a_zk. Historically, the concepts of scalar product and vector product were developed by Hamilton and Grassmann independently in the 1840’s. The word vector was first coined by Hamilton and it can be found in Hamilton’s (1853) works such as Lectures on Quaternions. The usual form of a quaternion is x = x₀ + x₁i + x₂j + x₃k, however, i.i = -1, j.j = -1, k.k = -1, i.j = 0, j.k = 0, and k.i = 0.

Questions for discussion:

1. How would you define a scalar product of two vectors algebraically?

2. How would you explain a scalar product of two vectors geometrically?

3. Why are scalar products of two vectors applicable in physics?

The moral of the lesson: a scalar product of two products can be defined algebraically and geometrically, and it can be applied in physics concepts such as work.

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.

3. Hamilton, W. R. (1853). Lectures on Quaternions. Hodges and Smith, Dublin.