ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

How to Solve for Properties and Proofs of the Unit Vector 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.

In our discussion on vectors, it is oftentimes helpful to look at the components of that vector. We can see those individual parts when we write out a vector as a = <a1, a2, a­3>. But what does each of these components mean? How can we find a way to look at the basic information that the parts mention? We must be careful not to get this confused with scalars, for those merely lengthen the vector or decrease it. We need a way to talk about the unit vector, or the most basic portion of the vector that conveys the information of it properly in a single length. But first, we need to examine how to find the length of any vector before we can talk about the basic material of that vector, and other fun properties from there.

Scalar multiples.
Scalar multiples.

The Length of a Scalar Multiple

When we take a vector and multiply it by a scalar, we are changing each of the components of the vector by the scalar. Whether the vector is increased or decreased in length and also changed in direction depends on the value of the scalar that is being applied to it. So how can we know how long that vector will be? Well, to find the length of a vector we need the magnitude, so for vector a that is increased by scalar c, the magnitude

||ca|| = [(ca1)2 + (ca2)2 + (ca3)2]0.5

= [c²a1² + c²a2² + c²a3²]0.5

= [c²(a1² + a2² + a3²)]0.5

Now, if I take the square-root of a number, then either the positive or negative value will work. A common example is (4)0.5. Either 2 or -2 squared gives us 4. So (c2)0.5 = c or –c. But since we want the length of the vector, we just care about the actual number and not the sign of it, so we need the absolute value of the square rooted number. So

[c²(a1² + a2² + a3²)]0.5 = |c|(a1² + a2² + a3²)0.5

=|c| ||a||

So if a vector is increased by a scalar, then the length of the vector is just the magnitude times the absolute value of the scalar, or ||ca|| = |c| ||a||. So we can see that the magnitude is a quantity we can solve for independent of the scalar. Does this help us in our discussion of the basic vector components? (Larson 766)

The Unit Vectors

Since the magnitude itself is independent of the scalar we apply to it, we can use that to find the unit vector, or the basic vector whose magnitude equals one. This is important because we will gain the ability to talk about the components of the vector in the x, y, and z directions and how much of the vector is in each of those directions. The goal of the unit vector is to have a magnitude (length) of 1. To achieve this, we need to normalize our vector a, or divide it by the length of the vector. Hence, the unit vector of a is a / ||a||. So how do we know that this new unit vector we have created will have a length of 1? Well, so long as a is not the zero vector, the magnitude of the unit vector is || a / ||a|| ||. But notice that

|| a / ||a|| || = |(1/||a||)|* (||a||)

Because from before, we know that any scalar does not change the magnitude of the vector, and the |(1/||a||) is just a scalar. Also, since the result of any absolute value is a positive number,

|(1/||a||)|* (||a||) = (1/||a||)* (||a||)

But now I have the same number on top and the same number in the denominator, so

(1/||a||)* (||a||) = 1.

So, by normalizing any vector, we will create a unit vector in the direction of a (766).


Notice that if vector a = <a1, a2, a3>, we can split this up as

<a1, a2, a3> = <a1, 0, 0> +<0, a2, 0> + <0, 0, a3>

Because of vector addition. Also notice that we can rewrite the vector as

<a1, 0, 0> +<0, a2, 0> + <0, 0, a3> = a1<1, 0, 0> + a2<0, 1, 0> + a3<0, 0 ,1>

Because of scalar multiplication. But notice that I now have a bunch of vectors that are 1.
These are all special unit vectors because of the direction they point in. We use the letters i, j, and k to represent these unit vectors. The unit vector i = <1, 0, 0>, j = <0, 1, 0>, and k = <0, 0, 1>. So we can rewrite

a = a1<1, 0, 0> + a2<0, 1, 0> + a3<0, 0 ,1> = a1i + a2j + a3k

The i component tells us how much of the vector is in the x-direction, the j component tells us the amount in the y-direction, and the k component tells us the amount in the z-direction. With these components known, it becomes easier to employ the properties of vectors and also give us a way to talk about how a vector operates in the x-y-z space, all with the basic components of each vector (767).

The basis for SOH-CAH-TOA.
The basis for SOH-CAH-TOA.

So what if I did not know that a = <a1, a2, a3> ? How could I get the x, y , and z components? Let’s look at an x-y example. If you know the number of degrees that the vector is above/below an axis, you can use SOH-CAH-TOA. In this case, you know that the hypotenuse is the length of the vector ||a|| but what would that opposite side, or the y-component, be? Sin Θ = O/H = O/||a|| so that opposite side will equal ||a|| Sin Θ. Similarly the adjacent side, or the x-component, will equal ||a|| Cos Θ. Therefore, for any vector that is Θ above the x-axis,

a = ||a|| Cos Θ i + ||a|| Sin Θ j (768).

Works Cited

Larson, Ron, Robert Hostetler, and Bruce H. Edwards. Calculus: Early Transcendental Functions. Maidenhead: McGraw-Hill Education, 2007. Print. 766-8.

© 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)