Physics Tutorial 2.4 - Dot (Scalar) Product of Two Vectors

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There are 2 lessons in this physics tutorial covering Dot (Scalar) Product of Two Vectors. The tutorial starts with an introduction to Dot (Scalar) Product of Two Vectors and is then followed with a list of the separate lessons, the tutorial is designed to be read in order but you can skip to a specific lesson or return to recover a specific physics lesson as required to build your physics knowledge of Dot (Scalar) Product of Two Vectors. you can access all the lessons from this tutorial below.

In addition to the tutorial for Dot (Scalar) Product of Two Vectors on this page, you can also access the following Vectors and Scalars learning resources for Dot (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 (F⃗) and Displacement (or linear distance) Δx⃗. However, there is another quantity known as Moment of Force (M⃗) 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 = F⃗ ∙ ∆x⃗

and

M⃗ = F⃗ ∙ ∆x⃗

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 a⃗ and b⃗ represents the numerical product of the magnitude of the vector a⃗ and the projection of the vector b⃗ in the direction of a⃗.

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

(b⃗_||a represents the component of the vector b⃗ in the direction of a⃗ and b⃗_⊥a the component of vector b⃗ perpendicular to a⃗)

Please select a specific "Dot (Scalar) Product of Two Vectors" lesson from the table below, review the video tutorial, print the revision notes or use the practice question to improve your knowledge of this physics topic.

Vectors and Scalars Learning Material
Tutorial ID	Physics Tutorial Title	Revision Notes	Revision Questions
2.4	Dot (Scalar) Product of Two Vectors
Lesson ID	Physics Lesson Title	Lesson	Video Lesson
2.4.1	How to calculate the dot (scalar) product of two vectors
2.4.2	Dot (scalar) product applications 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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Physics Tutorial 2.4 - Dot (Scalar) Product of Two Vectors

Introduction

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

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