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.

Section 11–6 Newton’s laws in vector notation

(Newton’s laws / Acceleration vector / Components of acceleration)

In this section, Feynman discusses Newton’s second law of dynamics, acceleration vector, and components of acceleration.

1. Newton’s laws:

“… we need not write three laws every time we write Newton’s equations or other laws of physics (Feynman et al., 1963, section 11–6 Newton’s laws in vector notation).”

This section could be titled as Newton’s second law of dynamics in vector notation. In a sense, Feynman seems quite sloppy when he uses the word law which means an equation. He explains the advantage of writing Newton’s laws as F = ma because this reduces the need of writing them as three laws that contain x’s, y’s, and z’s separately: F_x = m(dv_x/dt) = m(d²x/dt²) = ma_x, F_y = m(dv_y/dt) = m(d²y/dt²) = ma_y, F_z = m(dv_z/dt) = m(d²z/dt²) = ma_z. According to Feynman, what looks like one law is really three laws for any particular set of axes because any vector equation involves the statement that each of the components is equally important. Similarly, in section 9–3 “Components of velocity, acceleration, and force,” he mentions that there are really three laws with respect to the components of the force in the x-, y-, and z-direction.

Based on a definition of acceleration in vector notations, Newton’s second law of dynamics can be written as m(d²r/dt²) = F instead of three equations using Cartesian coordinates. Feynman explains that Newton’s second law as expressed in two-dimensional polar coordinates has the advantage that we need not always write Newton’s equations as three laws. However, physics teachers should elaborate that polar coordinates are useful for problems involving radial forces and in navigation either on the sea or in the air. On the other hand, one may prefer using Cartesian coordinates to solve problems if a system has little symmetry and thus, each direction can be considered separately. In short, the advantage or disadvantage of a coordinate system is dependent on whether we can solve a problem in a convenient manner.

2. Acceleration vector:

“… to write Newton’s laws in vector form, we have to go just one step further, and define the acceleration vector (Feynman et al., 1963, section 11–6 Newton’s laws in vector notation).”

The acceleration vector is the time derivative of the velocity vector, a = dv/dt. The components of this vector are the second derivatives of x, y, and z with respect to time: a_x = dv_x/dt = d²x/dt², a_y = dv_y/dt = d²y/dt², and a_z = dv_z/dt = d²z/dt². Feynman also mentions that acceleration is simply Δv/Δt. However, some physicists may disagree and prefer using the term average acceleration vector and instantaneous acceleration vector. To be precise, average acceleration vector, Δv/Δt, is the difference of the velocity vectors divided by a short time interval. Furthermore, instantaneous acceleration vector, dv/dt, is measured over an infinitely small interval (or an infinitesimal). This idealized concept is defined as the limit of average acceleration vector as the time interval approaches zero.

Feynman illustrates a misconception using a figure where a particle moves at a velocity v₁ at t = t₁ and velocity v₂ at t = t₂ a little later. The average acceleration vector is the difference of the velocity vectors (v₂ and v₁) divided by a short time interval. One should not subtract the initial velocity vector, v₁, by joining the ends of v₂ and v₁ as shown in figure 11–7 “A curved trajectory.” The correct difference of the velocity vectors, Dv, is shown in figure 11–8 “Diagram for calculating the acceleration” where the tails of the vectors are joined in the same location based on the law of vector subtraction. Interestingly, the instantaneous acceleration vector (dv/dt) and the force (F) are in the same direction, but this force is not really (or ontologically) the same as the acceleration vector despite having the equality sign in the equation.

3. Components of acceleration:

“…we can think of acceleration as having two components, Δv_∥, in the direction tangent to the path and Δv_⊥ at right angles to the path (Feynman et al., 1963, section 11–6 Newton’s laws in vector notation).”

The acceleration vector can be conceptualized as having two perpendicular components: tangential acceleration is in the direction tangential to a particle’s path and radial acceleration is perpendicular to the path. In general, the tangential acceleration (a_∥) is a measure of how fast an object’s tangential velocity changes. Feynman states that the acceleration tangential to the path is just the change in the length of the vector and it is the change in the speed v: a_∥ = dv/dt. Specifically, the tangential acceleration can be shown as a_∥ = r(d²θ/dt²) + 2(dr/dt)(dθ/dt). The term r(d²θ/dt²) is due to the object’s changing angular velocity (or angular acceleration). The other term 2(dr/dt)(dθ/dt) is known as the Coriolis acceleration that is due to the product of the object’s radial velocity and its angular velocity.

In his Tips on Physics, Feynman adds that “it’s good to remember that formula because it’s a pain in the neck to derive it: lal = v²/r (Feynman et al., 2006, p. 46).” Mathematically, the radial acceleration of an object can be derived in only three steps: a_⊥ = Δv/Δt = vΔθ/Δt = vw (or v²/r). Firstly, we deduce the difference of the velocity vectors to be Δv = vΔθ by assuming the object’s path to be approximately circular. Secondly, the radial acceleration a_⊥ = v(Δθ/Δt) if the magnitude of the velocity (v) is constant. Thirdly, if the radius of the path is approximately r, then we deduce the angular velocity (Δθ/Δt) to be equal to v/r and conclude that a_⊥= v²/r. To be more precise, we can derive the instantaneous radial acceleration (assuming the speed is not constant) as a_⊥= d²r/dt² – v²/r.

Questions for discussion:

1. Are there three laws in the equation, F = ma?

2. How would you define an acceleration vector?

3. How would you explain the components of acceleration in a curved path of a particle?

The moral of the lesson: to determine the acceleration vector, one should understand that the difference of two velocity vectors has meaning only when the tails of the vectors coincide in the same place.

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.