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  • Calculate the dot product of two given vectors.
  • Determine whether two given vectors are perpendicular.
  • Find the direction cosines of a given vector.
  • Explain what is meant by the vector projection of one vector onto another vector, and describe how to compute it.
  • Calculate the work done by a given force.

If we apply a force to an object so that the object moves, we say that work is done by the force. In Introduction to Applications of Integration on integration applications, we looked at a constant force and we assumed the force was applied in the direction of motion of the object. Under those conditions, work can be expressed as the product of the force acting on an object and the distance the object moves. In this chapter, however, we have seen that both force and the motion of an object can be represented by vectors.

In this section, we develop an operation called the dot product , which allows us to calculate work in the case when the force vector and the motion vector have different directions. The dot product essentially tells us how much of the force vector is applied in the direction of the motion vector. The dot product can also help us measure the angle formed by a pair of vectors and the position of a vector relative to the coordinate axes. It even provides a simple test to determine whether two vectors meet at a right angle.

The dot product and its properties

We have already learned how to add and subtract vectors. In this chapter, we investigate two types of vector multiplication. The first type of vector multiplication is called the dot product, based on the notation we use for it, and it is defined as follows:


The dot product of vectors u = u 1 , u 2 , u 3 and v = v 1 , v 2 , v 3 is given by the sum of the products of the components

u · v = u 1 v 1 + u 2 v 2 + u 3 v 3 .

Note that if u and v are two-dimensional vectors, we calculate the dot product in a similar fashion. Thus, if u = u 1 , u 2 and v = v 1 , v 2 , then

u · v = u 1 v 1 + u 2 v 2 .

When two vectors are combined under addition or subtraction, the result is a vector. When two vectors are combined using the dot product, the result is a scalar. For this reason, the dot product is often called the scalar product . It may also be called the inner product .

Calculating dot products

  1. Find the dot product of u = 3 , 5 , 2 and v = −1 , 3 , 0 .
  2. Find the scalar product of p = 10 i 4 j + 7 k and q = −2 i + j + 6 k .
  1. Substitute the vector components into the formula for the dot product:
    u · v = u 1 v 1 + u 2 v 2 + u 3 v 3 = 3 ( −1 ) + 5 ( 3 ) + 2 ( 0 ) = −3 + 15 + 0 = 12.
  2. The calculation is the same if the vectors are written using standard unit vectors. We still have three components for each vector to substitute into the formula for the dot product:
    p · q = u 1 v 1 + u 2 v 2 + u 3 v 3 = 10 ( −2 ) + ( −4 ) ( 1 ) + ( 7 ) ( 6 ) = −20 4 + 42 = 18.
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Find u · v , where u = 2 , 9 , −1 and v = −3 , 1 , −4 .


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Like vector addition and subtraction, the dot product has several algebraic properties. We prove three of these properties and leave the rest as exercises.

Properties of the dot product

Let u , v , and w be vectors, and let c be a scalar.

i. u · v = v · u Commutative property ii. u · ( v + w ) = u · v + u · w Distributive property iii. c ( u · v ) = ( c u ) · v = u · ( c v ) Associative property iv. v · v = v 2 Property of magnitude

Questions & Answers

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
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Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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Damian Reply
absolutely yes
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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Practice Key Terms 7

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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