5.1.1. The Scalar Product

The scalar product or dot product of any two vectors A and B, denoted as A.B (read A dot B) is defined as

A.B = A B cos θ (5.1a)

where θ is the angle between the two vectors. Since A, B and cos θ are scalars, the dot product of A and B is a scalar quantity. Each vector A and B has a direction, but their scalar product does not have a direction.

The scalar product, also known as the dot product, is a way to multiply two vectors that results in a scalar (a number without direction). The formula A.B = A B cos θ illustrates that it depends on the magnitudes of both vectors A and B as well as the cosine of the angle θ between them. This means that the scalar product is maximized when the two vectors point in the same direction (θ = 0°) and minimized (zero) when they are perpendicular (θ = 90°). Thus, the scalar product gives us information about both the lengths of the vectors and the cosine of the angle between them.

Think of the scalar product like a projection of one vector onto another. For example, if you're pushing a shopping cart at an angle, the scalar product helps determine how much of your effort (force) is actually used to move the cart forward versus how much is 'wasted' pushing sideways. If you're facing straight down the cart aisle (θ = 0°), all your force contributes to moving it forward. If you're pushing at a right angle (θ = 90°), none of it helps move the cart; it’s all sideways, where the cart won't go.

From Eq. (5.1a), we can express A.B as:

A.B = A (B cos θ) = B (A cos θ)

Geometrically, B cos θ is the projection of B onto A. So, A.B is the product of the magnitude of A and the component of B along A. Alternatively, it is the product of the magnitude of B and the component of A along B.

This statement explores how to visualize the scalar product using geometry. When we say B cos θ, we refer to the length of the vector B as projected onto the direction of vector A. Similarly, A cos θ would represent the projection of vector A onto vector B. This means that when calculating the dot product, we are interested in how one vector extends in the direction of the other. Understanding this visually aids in comprehending what the scalar product signifies, especially in contexts of work and energy.

Imagine you're shining a flashlight at an angle towards a wall. The light from the flashlight represents vector B, and the direction perpendicular to the wall represents vector A. How much light hits the wall (effective light) depends on the angle of the flashlight beam. The effective light illuminating the wall is analogous to the scalar product, where the flashlight's total strength gives you the power and the cosine of the angle tells you how much of that power is actually illuminating the wall.

Equation (5.1a) shows that the scalar product follows the commutative law:

A.B = B.A

The scalar product obeys the distributive law:

A. (B + C) = A.B + A.C

Furthermore, A. (λ B) = λ (A.B) where λ is a real number.

These properties indicate that the scalar product behaves nicely under various mathematical operations. Commutative means that the order of multiplication doesn't matter; for instance, A.B will always equal B.A. The distributive law allows for expansion when dealing with the sum of vectors, showing how the dot product can break down into parts. The scalar multiplication property shows how scaling a vector by a number scales the dot product by the same number.

Suppose you have several friends (vectors B and C) pushing a car (vector A). The total effort (scalar product) you all put in depends on how hard each person pushes. Whether you push first or your friend does (commutative property), or how much effort each additional friend contributes to pushing the car (distributive property) doesn't change the outcome. Similarly, if your friends push harder or softer (scaling), the total effort changes correspondingly.

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