ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

How to Solve for Properties and Proofs of the Triple Scalar Product of Vectors 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 improve it.

The parallelepiped
The parallelepiped

The triple scalar product (TSP) is a unique blend of the dot product and the cross product, two very important operators in mathematics. It has a similar origin as the cross product, which was found when looking at the area of a parallelogram contained within two vectors. The TSP differs however, for its derivation is found by taking the volume of a parallelepiped (a 3-D object whose faces are made up of parallelograms) set by three vectors that are not in the same plane.

If vectors b and c make up the base of the parallelepiped, then we would have a parallelogram and we know from an earlier article that the area of a parallelogram is the magnitude of the cross product between the vectors, or that

Area of the base = || b x c ||

Now, what is the height of the parallelepiped? Well, it will be however much of the a vector is vertical to the base vectors. But how can I know how much of the vector that will be? When I take the cross product, I get a vector that is orthogonal to both b anc c. This vector will be in the direction of a. If I know how much a projects onto b x c then I will know its component in the vertical direction and hence the height. Therefore,

Height = || projb x ca ||

Or that the height of the parallelepiped is the magnitutde of the projection of a onto b x c.

Now I can find the volume.

Volume = Area of base times the height

= || b x c || || projb x ca ||

But we know from an earlier article that projb x ca = (b x c) [a ∙ (b x c)] / (||b x c||2) so

|| b x c || || projb x ca || = || b x c || ||(b x c) [a ∙ (b x c)] / (||b x c||2) ||

But note that

||(b x c) [a ∙ (b x c)] / (||b x c||2) || = ||[(b x c)/||b x c||] [a ∙ (b x c)] / ||b x c|| ||

And (b x c)/||b x c|| is just a unit vector in the direction of b x c. Again, from an earlier article we know that the magnitude of a unit vector is 1 so we can simplify the volume equation to

|| b x c || ||[a ∙ (b x c)] / ||b x c|| || = ||b x c|| | [a ∙ (b x c)] / ||b x c|| |

We now write the second term with absolute value bars because in removing the unit vector all we have left is a scalar, and the magnitude of a scalar is just a scalar (just as [52]0.5 is simply 5) but all we want is the positive value since we are dealing with a volume. Hence the bars. Now notice that we have a ||b x c|| in the numerator and the denominator, so they cancel and all we are left with is

|a ∙ (b x c)|

We call this term inside the absolute value bars the triple scalar product and the volume is just the absolute value of this. Note that the TSP is not a vector but a scalar, like the dot product and unlike the cross product (Larson 795).

The Triple Scalar Product Revealed

So what does the TSP look like when I try to get a value for it? If we expand it, we shall find out.

a ∙ (b x c) = a ∙ [i(b2c3 – b­3c2) - j(b1c3 – b­3c1) + k(b1c2 – b­2c1)]

=(a1i + a2j + a3k) ∙ [i(b2c3 – b­3c2) - j(b1c3 – b­3c1) + k(b1c2 – b­2c1)]

=a1(b2c3 – b­3c2) - a2(b1c3 – b­3c1) + a3(b1c2 – b­2c1)

Which we will notice is the same thing as

| a1 a2 a3 |

| b1 b2 b3 |

| c1 c2 c3 |

Therefore, the TSP is simply the determinant of a matrix with components of a, b, and c (794).

Note that order is important. Simply switching two rows in the matrix will completely change the answer. In fact, by doing so you change the signs to the opposite operation. To see this, look at the TSP of b ∙ (a x c). This would have the following matrix that has two rows switched from before:

| b1 b2 b3 |

| a1 a2 a3 |

| c1 c2 c3 |

Then that determinant would be

b1(a2c3 - a­3c2) - b2(a1c3 - a­3c1) + b3(a1c2 - a­2c1) = b1a2c3 - b13c2 - b2a1c3 + b23c1 + b3a1c2 - b3a2c1

= b3a1c2 - b2a1c3 - b3a2c1 + b1a2c3 - b13c2 + b23c1

By the commutative property. Now If I pull out the common a terms then

b3a1c2 - b2a1c3 - b3a2c1 + b1a2c3 - b13c2 + b23c1 = a1(b3c2 - b2c3) - a2(b3c1 + b1c3) + a­3(b1c2 + b2c1)

= -a1(b2c3 - b3c2) + a2(b1c3 - b3c1) - a­3(b2c1 - b1c2)

= (-1) [a1(b2c3 – b­3c2) - a2(b1c3 – b­3c1) + a3(b1c2 – b­2c1)]

=(-1) [a ∙ (b x c)]

We can clearly see now that by switching rows we change the sign of our answer. But if I made two switches, then the answer would be (-1)(-1) times the original answer which would be the original answer. Therefore

a ∙ (b x c) = b ∙ (c x a)= c ∙ (a x b)

All of which involve two switches (794).

When the TSP Equals Zero

We used the TSP to find the volume of a parallelepiped earlier in this article, but one special condition that was not pointed out was that all three of the vectors must share a corner and they cannot be coplanar, or exist in the same plane. But what if they did? What would happen to the TSP? Well, if they were coplanar and in the same corner then they would all have to be the same vector, but possibly of different scalar lengths. Regardless, our matrix

| a1 a2 a3 |

| b1 b2 b3 |

| c1 c2 c3 |

Will still equal a1(b2c3 – b­3c2) - a2(b1c3 – b­3c1) + a3(b1c2 – b­2c1) but each component is just a scalar of the other. For our purposes, let’s say a = pb and c = fb. That is another way of showing how they are all related by different scalars. That will mean then that

a1(b2c3 – b­3c2) - a2(b1c3 – b­3c1) + a3(b1c2 – b­2c1) = pb1(b2fb3 – b­3fb2) - pb2(b1fb3 – b­3fb1) + pb3(b1fb2 – b­2fb1)

= pb1(0) - pb2(0) + pb3(0)

= 0

Therefore, if the vectors are coplanar, then the TSP will equal 0 (795).

Works Cited

Larson, Ron, Robert Hostetler, and Bruce H. Edwards. Calculus: Early Transcendental Functions. Maidenhead: McGraw-Hill Education, 2007. Print. 794-5.

© 2014 Leonard Kelley

Comments

    0 of 8192 characters used
    Post Comment

    No comments yet.

    working

    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, hubpages.com 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: https://hubpages.com/privacy-policy#gdpr

    Show Details
    Necessary
    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 googleapis.com or gstatic.com domains, for performance and efficiency reasons. (Privacy Policy)
    Features
    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)
    Marketing
    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.
    Statistics
    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)