Dot (Scalar) Product of Two Vectors | iCalculator™

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2.4Dot (Scalar) Product of Two Vectors

In this Physics tutorial, you will learn:

  • The meaning of "dot (scalar) product" of two vectors (both geometrically and conceptually)
  • How to calculate the dot (scalar) product of two vectors? (two methods)
  • Some applications of dot (scalar) product in Physics
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Introduction

In Physics, there are many quantities that apparently are measured by multiplying the same vector quantities (and they seemingly are measured by the same unit), but which describe concepts that are totally unrelated. For example, Work (W) is calculated by multiplying two vector quantities such as Force (F) and Displacement (or linear distance) Δx. However, there is another quantity known as Moment of Force (M) that is also calculated by multiplying Force and the linear distance from the application point (the position in which the force acts) to the turning point (we will discuss the concept of "Work" in Section 5 and that of "Moment of Force" in Section 6). The equations for these two quantities are apparently the same:

W = F ∙ ∆x

and

M = F ∙ ∆x

Since we have two different quantities obtained by multiplying the same vectors (force and linear distance), it is obvious that the two vector multiplication procedures used to calculate the Work and Moment of Force are different. This difference is outlined at the symbols used to express these multiplications. In the first case, the dot ( ∙ ) symbol is used to express the two vectors multiplication operation while in the second case, it is used the cross ( × ) symbol to represent the multiplication of the given vectors. This is not unintentional; rather, the two different symbols are used to demonstrate that here we are using two different techniques to find the product of two vectors. The first technique is known as the "dot (scalar) product of two vectors" and the second one is known as "cross (vector) product of two vectors".

In this Physics Tutorial, we will deal only with "dot (scalar)" product of two vectors. The cross (vector) product will be discussed in the next tutorial "Cross (vector) Product of Two Vectors".

The meaning of "dot (scalar) product" of two vectors

Geometrically, the dot (scalar) product of two vectors a and b represents the numerical product of the magnitude of the vector a and the projection of the vector b in the direction of a.

Look at the figure below:

Physics Tutorials: This image shows a vector with numerous points to illustrate the meaning of the dot scalar as part of the information below

(b||a represents the component of the vector b in the direction of a and b⊥a the component of vector b perpendicular to a)

How to calculate the dot (scalar) product of two vectors

If we appoint a basic direction (for example Ox) to the first vector a, we write as bx instead of b||a the component of the vector b in the direction of a. It is obvious the component of vector b perpendicular to a is denoted as by instead of b⊥a.

Also, we can appoint a letter (for example θ) to the acute angle formed by the vector a (or its extension) and the vector b. Therefore, the above figure is written as:

Physics Tutorials: This image shows 2 vectors to help illustrate how to calculate a dot scalar of two products

The component by is equal in length to the opposite side to the angle θ for the triangle shown in the figure. Therefore, we can "close" this triangle to form a right triangle as shown below.

Physics Tutorials: This image contains a rounded corner box placed on a surface to illustrate the calculation of mechanical power in physics

From trigonometry, it is known that in a right triangle (as the one shown in the above figure), we have

bx = |b| ∙ cos θ

and

by = |b| ∙ sin θ

Therefore, for the dot (scalar) product of the vectors a and b we can write:

a ∙ b = |a | ∙ |bx |
=|a | ∙ |b | ∙ cos θ

It is obvious the above product represents a scalar (number) c, not a vector, because we found it by multiplying two vector magnitudes (which are simply numbers) and the cosine of an angle (which is a number as well). Hence, if we denote by c the dot product of two vectors a and b, we obtain

c = |a| ∙ |b| ∙ cos θ

Example 1

Mechanical Power P in Physics is alternatively defined as the dot product of force F exerted on an object and the velocity v gained by the object due to the action of this force.

A 120 N force acts at an object at 200 to the horizontal direction as shown in the figure. As a result, the object moves at constant velocity by a magnitude of 0.4 m/s. Calculate the useful power delivered by the source in Watts (1 Watt = 1 Joule ∙ second). Take cos 200 = 0.94, sin 200 = 0.34.

Physics Tutorials: This image contains a grid with vectors u and v identified to support the physics tutorial on dot vectors

Solution 1

As stated above, by definition, Power is the scalar product of Force and Velocity. From the mathematical definition of the scalar product of two vectors, we know that to find the dot product of two vectors, we must multiply one vector with the component of the other vector in the direction of the first one. In the specific case, the movement occurs in the direction of the velocity vector. Therefore, we take the velocity vector as the first one and express the direction of force vector in terms of it. Thus, we must multiply in scalar mode the velocity vector v and the component of force F according to the direction of v. This means we have to calculate:

P = v ∙ F
=|v | ∙ |Fn the direction of v |
=|v | ∙ F ∙ cos θ

After substituting the values, we obtain

P = 0.4 m/s ∙ 120N ∙ 0.94
= 45.12 Watt

Another method for calculating the dot product of two vectors. Calculating the dot product using coordinates

If the coordinates of the two vectors are known, it is much easier to calculate their dot product. We don't need to know the angle in-between or the magnitudes of the vectors. We can use the formula:

a ∙ b = xa ∙ xb + ya ∙ yb

This formula is particularly useful when none of vectors lies according to a main direction (axis). Thus, instead of trying to calculate the angle in-between by finding the difference of two angles (the angle formed by the vector u to the horizontal axis minus the angle formed by the vector v to the horizontal axis) (look at the figure below), and then expressing the direction of the second vector in terms of the first one, it is much easier multiplying only the coordinates as no angle is needed here.

Physics Tutorials: This image contains a grid with vectors u and v identified to support the physics tutorial on dot vectors

For simplicity, both vectors are taken with their origin (tail) at (0, 0). The procedure to calculate the angle between the two vectors is quite long. First, we need to find the angles formed by each vector to the horizontal direction by considering the tangents (y-coordinate / x-coordinate). After finding each tangent, we find the respective angles using a scientific calculator. Afterwards, we subtract the angles to find the angle between the two vectors. Finally, we can use the (first) dot product formula to find the result. As you see, it is not worth trying it.

By using the new formula of dot product, we find the result in a much easier way. Thus, from the figure you can find all coordinates you need. They are: xu = 7 units, xv = 13 units, yu = 10 units and yv = 4 units.

Substituting the above values, we obtain

u ∙ v = xu ∙ xv + yu ∙ yv
= (7 ∙ 13 + 10 ∙ 4) units
= (91 + 40) units
= 131 units

Dot (scalar) product applications in Physics

There are many applications of dot (scalar) product of two vectors in Physics. Below, we will mention only a few of them.

1. Work as a dot product of Force and Displacement.

We have discussed earlier (in Introduction) this point. If you multiply in scalar mode the force exerted by an object and the displacement of the object due to the action of this force, the result will be a scalar quantity known as Work and measured in Joules. It is meaningless to say "the work done by the force F is 300 J due East" for example. We simply say "the work done by the force F is 300 J" as work is a quantity related to the change in energy of a system (see the article 5.1 "Work and Energy" for more detail); it does not involve any direction.

2. Power as a dot product of Force and Velocity.

We discussed this point earlier in the solved example. If we multiply in scalar mode the two vector quantities, Force and Displacement, we obtain a scalar result that is Power. It does not involve any direction as well.

3. The angle between two forces acting on the same object.

Let's consider again the solved example in the previous section (let's take the abovementioned vectors as force vectors and the units as newtons).

Physics Tutorials: This image contains a grid with vectors u and v identified to support the physics tutorial on dot vectors

We found the dot product of vectors u and v and it was 131 units. Now, let's calculate the magnitudes of each vector. From the article "Vectors and Scalars in Physics" we know that

|u| = √u2x + u2y = √72 + 102 = √49 + 100 = √149 units

and

|v| = √v2x + v2y = √132 + 42 = √169 + 16 = √185 units

Therefore, we obtain for the angle between u and v

u ∙ v = |u| ∙ |v| ∙ cos (∡ u,v)
cos (∡ u,v) = u ∙ v/|u| ∙ |v|
= 131/149 ∙ √185
= 131/166
= 0.789

Therefore, the angle between the two vectors u and v is cos-1 0.789 = 37.890.

This result was obtained much easier than if we used the (long) method described in the previous section.

4. Kinetic Energy as a dot product of linear momentum and velocity.

Kinetic Energy of a moving object (a scalar quantity measured in Joules like Work) is calculated through the equation

KE = 1/2 ∙ m ∙ v2

where m is the mass of the object and v is its velocity (look at the article "Kinetic Energy. Work-Kinetic Energy Theorem" for more info regarding this concept).

However, since the linear momentum of a moving object is p = m ∙ v (look at the article "Momentum and Impulse" for more info regarding this concept), we obtain for the kinetic energy KE

KE = 1/2 ∙ (m ∙ v ) ∙ v
= 1/2 ∙ p ∙ v

Therefore, Kinetic Energy can be expressed as the dot product of two vectors, p and v multiplied by a constant 1/2 (which only changes the result; it does not breaks the abovementioned rule).

These were some examples showing the most important applications of dot product in Physics.

We hope you found this tutorial useful, if you did. Please take the time to rate this tutorial and/or share on your favourite social network. In our next tutorial, we explore Cross (vector) Product of Two Vectors.

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