ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

How to Solve for Properties and Proofs of the Dot Product for Calculus

Updated on February 24, 2016
1701TheOriginal profile image

Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly explore it.

The Law of Cosines applied to vectors.
The Law of Cosines applied to vectors.

The Definition of the Dot Product

The dot product is an incredible tool of higher-level math, though many may not know how we formally arrive at its results but just know its applications. While it is good to know how tu use the properties, it is often insightful to know where they came from. To see this at work, observe two vectors and the difference between them. To find the hypotenuse, or the difference between vectors, we need to take advantage of the Law of Cosines, or that

c2 = a2 + b2 – 2abCos Θ.

In our case, we need to use the lengths of the vectors for our sides of the triangle, or ||a||, ||b||, and ||a-b||. So

||a-b||2 = ||a||2 + ||b||2 – 2 ||a||||b||Cos Θ.

Notice that

||a-b||2 = ([(a1 – b1)2 + (a2 – b2)2 + (a3 – b3)2]0.5)2

= (a1 – b1)2 + (a2 – b2)2 + (a3 – b3)2

= (a1 – b1)(a1 – b1) + (a2 – b2)(a2 – b2)+ (a3 – b3)(a3 – b3)

= a12 – 2a­1b1 + b­12 + a22 – 2a­2b2 + b­22 + a32 – 2a­3b3 + b­32

Putting this back in for ||a-b||2 means that

a12 – 2a­1b1 + b­12 + a22 – 2a­2b2 + b­22 + a32 – 2a­3b3 + b­32 = ||a||2 + ||b||2 – 2 ||a||||b||Cos Θ

But notice that

||a||2 + ||b||2 = [(a12 + a22 + a32)0.5]2 + [(b­12+ b­22 + b­32)0.5]2

= a12 + a22 + a32 + b­12+ b­22 + b­32


a12 – 2a­1b1 + b­12 + a22 – 2a­2b2 + b­22 + a32 – 2a­3b3 + b­32 = a12 + a22 + a32 + b­12+ b­22 + b­32 – 2 ||a||||b||Cos Θ

Wow, quite a lot there. But notice how we can cancel out all the squared terms from both sides! When we finish with this, we arrive at

– 2a­1b1 – 2a­2b2 – 2a­3b3 = – 2 ||a||||b||Cos Θ

We can simplify the -2 out of the left hand side and then divide both sides by it, giving us

a­1b1 + a­2b2 + a­3b3 =||a||||b||Cos Θ

So what is all that stuff on the left? We define that as the dot product and is known as a ∙ b. So we now know that

a ∙ b = a­1b1 + a­2b2 + a­3b3


a ∙ b = ||a||||b||Cos Θ

This also gives us another way to find Cos Θ, for by moving terms to the other side,

Cos Θ = (a ∙ b) / (||a||||b||)

We have a new way to find the angle between vectors (Larson 782). Now let’s see what else we can use the dot product for.

Additive Properties

It is important to note that the final result of the dot product is a number without direction, or a scalar. If you were to end up with a vector as your final answer then you know something is wrong and it is best to look over your work. It is also worth knowing if any of the previous properties of vectors apply to the dot product.

Does the commutative property apply? That is, does a ∙ b = b ∙ a? Well,

a ∙ b = a1b1 + a­2b2 + a3b3

But because the components are real numbers and the order that I multiply real numbers does not matter,

a1b1 + a­2b2 + a3b3 = b1a1 + b2a2 + b3a3

= b ∙ a

Yes, the dot product is commutative (781)

How about the distributive property? Will a ∙ (b + c) = a ∙ b + a ∙ c?

a ∙ (b + c) = a ∙ <(b1 + c1), (b2 + c­­2), (b3 + c3)>

= a1(b1 + c1) + a2(b2 + c­­2) + a3(b3 + c3)

But since I am multiplying a real number across a sum and the multiplication can be spread out,

a1(b1 + c1) + a2(b2 + c­­2) + a3(b3 + c3) = a1b1 + a1c1 + a2b2 + a2c­­2 + a3b3 + a­3c3

And because the order that I add real numbers doesn’t matter,

a1b1 + a1c1 + a2b2 + a2c­­2 + a3b3 + a­3c3 = a1b1 + a2b2 + a3b3 + a1c1 + a2c­­2 + a­3c3

= a ∙ b + a ∙ c

Yes, the dot product can be distributed (781).

Multiplicative Properties

What happens if I try to distribute a scalar across the dot product? Does c(a ∙ b) = ca ∙ cb? It doesn’t. Let’s see why.

c(a ∙ b) = c(a1b1 + a­2b2 + a3b3)

= ca1b1 + ca­2b2 + ca3b3

= (ca1)b1 + (ca­2)b2 + (ca3)b3 OR = a1(cb1) + a­2(cb2)+ a3(cb3)

Because of the associative property. Therefore,

(ca1)b1 + (ca­2)b2 + (ca3)b3 = (ca) ∙ b


a1(cb1) + a­2(cb2)+ a3(cb3) = a ∙ (cb)

Indeed, the dot product is not a direct summation but the sum of products, so you cannot distribute as we normally would. The dot product c(a ∙ b) = (ca) ∙ b = a ∙ (cb) (781).

The dot product of any vector and 0 is equal to 0. Note that it equals the number 0 and not the vector. To see this,

a ∙ 0 = a10 + a20 + a30

= 0 + 0 + 0 = 0 (781).

What does the dot product of a vector and itself equal?

a ∙ a = a1a1 + a2a2 + a3a3

= a12 + a22 + a32

Which is the length of the vector squared, so

a12 + a22 + a32 = ||a||2

Therefore, a ∙ a = ||a||2

Orthogonal Vectors

Another important feature of the dot product tells us if two vectors are orthogonal, or perpendicular. Remember back to the unit circle for a moment. When we were at a 90 degree angle, the cosine (or x-component) was equal to zero. Therefore, if two vectors are orthogonal,

Cos 90 = (a ∙ b) / (||a||||b||) = 0


a ∙ b = 0

So if the dot product of two vectors equals zero, then you know that they are orthogonal (783).

Works Cited

Larson, Ron, Robert Hostetler, and Bruce H. Edwards. Calculus: Early Transcendental Functions. Maidenhead: McGraw-Hill Education, 2007. Print. 781-3.

© 2014 Leonard Kelley


    0 of 8192 characters used
    Post Comment

    No comments yet.


    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at:

    Show Details
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the or domains, for performance and efficiency reasons. (Privacy Policy)
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)
    ClickscoThis is a data management platform studying reader behavior (Privacy Policy)