<< Chapter < Page Chapter >> Page >
By the end of this section, you will be able to:
  • Explain the difference between the scalar product and the vector product of two vectors.
  • Determine the scalar product of two vectors.
  • Determine the vector product of two vectors.
  • Describe how the products of vectors are used in physics.

A vector can be multiplied by another vector but may not be divided by another vector. There are two kinds of products of vectors used broadly in physics and engineering. One kind of multiplication is a scalar multiplication of two vectors . Taking a scalar product of two vectors results in a number (a scalar), as its name indicates. Scalar products are used to define work and energy relations. For example, the work that a force (a vector) performs on an object while causing its displacement (a vector) is defined as a scalar product of the force vector with the displacement vector. A quite different kind of multiplication is a vector multiplication of vectors . Taking a vector product of two vectors returns as a result a vector, as its name suggests. Vector products are used to define other derived vector quantities. For example, in describing rotations, a vector quantity called torque is defined as a vector product of an applied force (a vector) and its distance from pivot to force (a vector). It is important to distinguish between these two kinds of vector multiplications because the scalar product is a scalar quantity and a vector product is a vector quantity.

The scalar product of two vectors (the dot product)

Scalar multiplication of two vectors yields a scalar product.

Scalar product (dot product)

The scalar product     A · B of two vectors A and B is a number defined by the equation

A · B = A B cos φ ,

where φ is the angle between the vectors (shown in [link] ). The scalar product is also called the dot product    because of the dot notation that indicates it.

In the definition of the dot product, the direction of angle φ does not matter, and φ can be measured from either of the two vectors to the other because cos φ = cos ( φ ) = cos ( 2 π φ ) . The dot product is a negative number when 90 ° < φ 180 ° and is a positive number when 0 ° φ < 90 ° . Moreover, the dot product of two parallel vectors is A · B = A B cos 0 ° = A B , and the dot product of two antiparallel vectors is A · B = A B cos 180 ° = A B . The scalar product of two orthogonal vectors vanishes: A · B = A B cos 90 ° = 0 . The scalar product of a vector with itself is the square of its magnitude:

A 2 A · A = A A cos 0 ° = A 2 .
Figure a: vectors A and B are shown tail to tail. A is longer than B. The angle between them is phi. Figure b: Vector B is extended using a dashed line and another dashed line is drawn from the head of A to the extension of B, perpendicular to B. A sub perpendicular is equal to A magnitude times cosine phi and is the distance from the vertex where the tails of A and B meet to the location where the perpendicular from A to B meets the extension of B. Figure c: A dashed line is drawn from the head of B to A, perpendicular to A. The distance from the tails of A and B to where the dashed line meets B is B sub perpendicular and is equal to magnitude B times cosine phi.
The scalar product of two vectors. (a) The angle between the two vectors. (b) The orthogonal projection A of vector A onto the direction of vector B . (c) The orthogonal projection B of vector B onto the direction of vector A .

The scalar product

For the vectors shown in [link] , find the scalar product A · F .

Strategy

From [link] , the magnitudes of vectors A and F are A = 10.0 and F = 20.0. Angle θ , between them, is the difference: θ = φ α = 110 ° 35 ° = 75 ° . Substituting these values into [link] gives the scalar product.

Solution

A straightforward calculation gives us

A · F = A F cos θ = ( 10.0 ) ( 20.0 ) cos 75 ° = 51.76 .
Got questions? Get instant answers now!

Check Your Understanding For the vectors given in [link] , find the scalar products A · B and F · C .

A · B = −57.3 , F · C = 27.8

Got questions? Get instant answers now!

Questions & Answers

how does Neisseria cause meningitis
Nyibol Reply
what is microbiologist
Muhammad Reply
what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
Ntefuni Reply
What is microbiology
Mercy Reply
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
Ziyad Reply
How bacteria create energy to survive?
Muhamad Reply
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
what is sporadic nd endemic, epidemic
Aminu Reply
the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
explain assimilatory nitrate reduction
Esinniobiwa Reply
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Shu Reply
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Micheal Reply
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
what is cell
faisal Reply
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
Akanni
ok
Innocent
cell is the structural and functional unit of life
Hasan
is the fundamental units of Life
Musa
what are emergency diseases
Micheal Reply
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
Lubega
Heyy Lubega hussein where are u from?
_Adnan
en français
Adama
which site have a normal flora
ESTHER Reply
Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
Safaa
skin
Asiina
skin,Oral,Nasal,GIt
Sadik
How can Commensal can Bacteria change into pathogen?
Sadik
How can Commensal Bacteria change into pathogen?
Sadik
all
Tesfaye
by fussion
Asiina
what are the advantages of normal Flora to the host
Micheal
what are the ways of control and prevention of nosocomial infection in the hospital
Micheal
what is inflammation
Shelly Reply
part of a tissue or an organ being wounded or bruised.
Wilfred
what term is used to name and classify microorganisms?
Micheal Reply
Binomial nomenclature
adeolu
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 6

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 1' conversation and receive update notifications?

Ask