L12 lines and planes in space

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Lines and planes in space. Upcoming SlideShare. Like this document? Why not share! Embed Size px. Start on. Show related SlideShares at end. WordPress Shortcode. Full Name Comment goes here. Are you sure you want to Yes No. Be the first to like this. No Downloads. Views Total views. Actions Shares. No notes for slide. Lines and planes in space 1. Such axes can be used to describe points in 3-space by triples of numbers: the signed distances to planes formed by two of these axes.

We call such a set of perpendicular axes through a given point a cartesian coordinate system. Each of the axes is called a coordinate axis and the point of intersection of the three axes is called the origin. Every point of a coordinate axis cor- responds to a real number.

As you know, often, but not necessarily always, these axes are called x—axis, y—axis and z—axis. By convention, we agree that the coordinate sys- tem be right—handed : if you turn the positive x—axis to the positive y—axis, then a screw positioned along the z—axis following this movement would move in the direction of the positive z—axis. Every two of the three axes span a plane: the y, z—plane, the x, z—plane and the x, y—plane, respectively. The signed distances u, v and w of a point P in 3—space to each of these planes in the given order are the coordinates of P ; we usually put the three together as follows: u, v, w and still call this the coordinates of P.

Notations for points that occur often: x, y, zx1x2x3a, b, ca1a2a3u, v, wetc. Right: Cartesian coordinate systems are taken to be right—handed: a screw positioned along the z—axis moves along the positive z—axis if turned in the direction from x—axis to y—axis. Most discussions in this chapter will focus on 3—space.By now, we are familiar with writing equations that describe a line in two dimensions.

To write an equation for a line, we must know two points on the line, or we must know the direction of the line and at least one point through which the line passes. In two dimensions, we use the concept of slope to describe the orientation, or direction, of a line. In three dimensions, we describe the direction of a line using a vector parallel to the line.

In this section, we examine how to use equations to describe lines and planes in space. Recall that parallel vectors must have the same or opposite directions.

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Note that the converse holds as well. This set of three equations forms a set of parametric equations of a line:. We can set them equal to each other to create symmetric equations of a line:. The parametric equations of a line are not unique.

12.5: Equations of Lines and Planes in Space

Using a different parallel vector or a different point on the line leads to a different, equivalent representation. Each set of parametric equations leads to a related set of symmetric equations, so it follows that a symmetric equation of a line is not unique either.

We have. We already know how to calculate the distance between two points in space. We now expand this definition to describe the distance between a point and a line in space. Several real-world contexts exist when it is important to be able to calculate these distances. Air travel offers another example.

l12 lines and planes in space

Airlines are concerned about the distances between populated areas and proposed flight paths. We still define the distance as the length of the perpendicular line segment connecting the point to the line.

In space, however, there is no clear way to know which point on the line creates such a perpendicular line segment, so we select an arbitrary point on the line and use properties of vectors to calculate the distance. Using a formula from geometry, the area of this parallelogram can also be calculated as the product of its base and height:.

We can use this formula to find a general formula for the distance between a line in space and any point not on the line. Given two lines in the two-dimensional plane, the lines are equal, they are parallel but not equal, or they intersect in a single point. In three dimensions, a fourth case is possible. For each pair of lines, determine whether the lines are equal, parallel but not equal, skew, or intersecting.

Because the direction vectors are not parallel vectors, the lines are either intersecting or skew.

To find this point, we use the parametric equations to create a system of equalities:. Substitution into the third equation, however, yields a contradiction:. These lines do not intersect, so they are skew see the following figure. We know that a line is determined by two points. In other words, for any two distinct points, there is exactly one line that passes through those points, whether in two dimensions or three. Similarly, given any three points that do not all lie on the same line, there is a unique plane that passes through these points.

Just as a line is determined by two points, a plane is determined by three. This may be the simplest way to characterize a plane, but we can use other descriptions as well. For example, given two distinct, intersecting lines, there is exactly one plane containing both lines. A plane is also determined by a line and any point that does not lie on the line. These characterizations arise naturally from the idea that a plane is determined by three points.

Perhaps the most surprising characterization of a plane is actually the most useful. Imagine a pair of orthogonal vectors that share an initial point.Or, search for the description of the feature you want to use from the Help menu in Preview. When you're done editing a file, you can save it as a new file. Choose Export from the File menu to save the file in a new format.

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Select the format you want to use from the Format pop-up menu that appears in the save sheet. You can also share files directly from Preview by using the Share button in the toolbar of the Preview window. This makes it easy to quickly email or chat a file, or add images to Photos. You can also share files using any social network accounts you've set up in Internet Accounts. Open Menu Close Menu Apple Shopping Bag Apple Mac iPad iPhone Watch TV Music Support Search support.

View images and PDFsYou can open an image using Preview by double-clicking the image or dragging it to the Preview icon in the Finder or Dock. Edit imagesTo edit an image, click the Edit button from the toolbar at the top of the Preview window. Selection toolYou can use the selection tool to grab a portion of an image that you want to copy from a document. For Rectangular selection and Elliptical selection, drag your pointer across the image to select part of it.

Hold the Shift key while dragging to constrain your selection to a square or circle. The Lasso tool allows you to outline a selection by dragging your pointer in a free form around the image. The Smart Lasso tool lets you create a free form selection that "snaps" to elements of the image that appear to be a similar color. Instant alphaFor some image types, such as PNG, you can use the Instant Alpha tool to select and remove a background or other object from an image. Shape toolsUse the RectangleEllipse or Line tools to add shapes to your image.

l12 lines and planes in space

Text toolUse the Text tool to add text to your image. Adjust colorClick Adjust Color button to open the Adjust Color window. Adjust sizeYou can quickly scale an image by clicking the Adjust Size button in the toolbar, or from the Tools menu. Edit PDF filesPreview opens PDF (Portable Document Format) files so that you can view, edit and share them. Select text Use this tool to select text in a PDF document.

Add a shape Use these tools to insert a box, oval or line in your document. Select an area Use this tool to select an image or other non-text part of your document, including any shapes or other annotations you have added to the PDF.

Add text Select this tool, then drag your pointer across a document to create a text box you can type in. Notes Use this tool to create annotations you can embed in your PDF. Signature Use this tool to insert your signature into a PDF document. Fill in formsIf you receive a form in PDF format, you can easily fill in form blanks by clicking or double-clicking each form field.

If you are unable to fill in a form by clicking it, you can also use the Add Text tool to place an editable text box onto a formSign documentsPreview allows you to add a digital image of your signature to PDF documents.

Click the Edit button in the Preview window so that the editing toolbar is displayed. Click the Signature button in the editing toolbar. Choose the option to create a signature. Sign your name to a piece of paper, then hold that paper up to the camera on your Mac. Move the paper so that the signature is drawn along the line that appears on screen. Click Accept to save the signature.First off your business page must have at least 1 review.

Assuming it does this has worked for me for a handful of sites. Open google and search for your business using the full address. When your business page appears on the right hand side, click on the view review button.

When the review box opens copy the full URL and use this for your link. When opened by a PC it shows your reviews with a post review button at the top. When you view on a mobile ( I have checked Firefox and Chrome on android and Safari on an Iphone) the review box will open. Above the posted reviews are the outlined 5 stars. The contents of this blog post may not work for everyone.

When I get a little free time, I hope to find a better solution. If you find one before I do, please post it here. This is why I focus on Google reviews over reviews on any other website.

Google uses reviews as a ranking factor in search results (at least locally), yet makes it difficult for customers to review a business. It just reloads the same page. It sounds like you set your page up incorrectly at the beginning. Your business does not have a physical address attached to it. If you want to get reviews, you must add your address to your business page. How do I add a physical address to my Google page.

What is the link to your Google Business page. I will take a lookYes, I have the same problem. I think the cool icons are the icons at the left-hand side such as review, directions, star, upload photo. My page only has share. The funny thing is I have another page that does have the icons, and the review box does pop up.

l12 lines and planes in space

If I make a version of your link with my Google page (to go right to the review), NOTHING happens. You just get my page. I have been into Settings. Nice write up Travis, I was just figuring this out the other day after Google slammed Local and thus shut down our old approach. Unfortunately from what I can tell, going to your Google page thru google. For my business, I have a few dozen interactions with clients or potential clients on my absolute busiest days.

Will google not let them post because it would appear you are fishing for reviews.

l12 lines and planes in space

For those following this article, I have found a way to include a direct link for leaving a mobile review. I am still working out a few of the details, but I hope to have it complete soon. I tried different links to Business Google pages, but this is HANDS DOWN the most direct and best method. Why is Google so ignorant not to place these instructions in their help pages.

Do you know of a fix for this. Hmm Without being able to duplicate it, I cannot trouble-shoot it. My guess is that the issue lies on your computer with a improper clearing of the cache or something else similar. Sorry to give such basic suggestions, but as I mentioned above, I cannot troubleshoot the issue very well, if I cannot replicate the issue.Tip-off is set for.

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Fordham starts fast, beats St. Francis (BKN) 76-68 NEW YORK (AP) Will Tavares had 21 points, Joseph Chartouny added 19 and Prokop Slanina dropped in 18 as Fordham jumped out to an early lead and then held off St. Francis (Brooklyn) for a 76-68 win. Team bankroll is running down the betting odds for Week 16 EPL action and giving you free picks and football betting predictions galore.

Former four division world champion Miguel Cotto (41-5, 33 KOs) steps inside the squared circle for the final time in his Hall of Fame career as he defends the WBO junior middleweight title Saturday against 2008 Olympian Sadam Ali (25-1, 14 KOs) from Madison Square Garden in New York City (HBO 10 p.

Cotto won the title in his last fight in August after a 21-month layoff when he defeated Yoshihiro Kamegai by unanimous decision. We will find out whether Cotto has one eye on retirement and a foot out the door if he does, Ali can spoil the party inside MSG and pull off a monumental upset.

MORE: Cotto on how he wants to be rememberedWhen boxing fans get a fight with a big name like Cotto, there are sure to be plenty of bets made. Ali odds are pretty wide considering the where the two combatants are at in their careers. Online sportsbook Bovada's fight odds (as of Dec.

This isn't the fight Cotto was looking for. He attended the Sept. Cotto and Golden Boy Promotions made overtures to some of the bigger names in the sport, including welterweight champion Errol Spence Jr. After those options were exhausted, the offer was extended to Ali.

The 29-year-old heads into the biggest fight of his career, riding a three-fight winning streak since his knockout loss to Vargas in March 2016. His most recent fight was July 29 against Johan Perez on a "Golden Boy Boxing on ESPN" card, winning by unanimous decision. While anything can happen once Cotto and Ali step into the ring, the consensus in boxing is this will be a Cotto rout.

Ali's never faced anyone in the same league as Cotto, and he needs to have a once-in-a-lifetime performance to be able to pull off the upset.

Cotto gets the job done. He's the better pure boxer, has more power and fight experience, as this will mark the 10th time he's competed inside MSG.

Add it up and it equals a successful swan song for the one of the all-time greats.The material on this site may not be reproduced, distributed, transmitted, cached or otherwise used, except with the prior written permission of Alabama Media Group.

The picks with the winning team in bold at the top of each prediction: Champion: Georgia "I'm thinking heavy on Georgia because they're a hard fighting team and they have a bunch of guys on that team that can make plays when it's called their way. Oklahoma "One reason: Baker Mayfield. Clemson "They're ranked first in defensive efficiency and their defensive line is unbelievable. Clemson "Well my brother is a linebacker at Clemson, so I have to go with them, for his sake.

Georgia "Georgia, all the way. View Comments About Us About Alabama Media Group Jobs at Alabama Media Group Advertise with us News In Education Frequently Asked Questions About AL. Manage your subscription Delivery feedback Place a vacation hold Make a payment Customer Service Place an ad Get home delivery Promote your event in our calendar Send us an email Submit a news tip Buy newspaper front pages, poster and more AL.

Move cursor over product parameter name to display the graphic -- click to enlarge. Links to these same products are also available below. Hazards Outlook 3-7 Day 8-14 Day 8-14 Day Temps U. In this space, however, as the work week comes to an end we'll crank up the heat a bit more than usual and go off the map. Every week, our analysts will drop their boldest predictions and inevitably dance around their desks with idiotic grins if they ever happen to come true.

If said predictions go belly up, we shall never speak of them again. Alex Gelhar: Eli Manning scores more fantasy points than Dak PrescottSure, this sounds impossible, considering the fact that Eli Manning has more single-digit fantasy point outings (five) than games with 20-plus points (two).

But something tells me this is one of those weeks for the Giants. Manning, the face of their franchise for the last decade-plus is back under center. And the Cowboys, a team with more holes than the plot of "Jurassic World," are their opponents. I'm not buying Dallas' win last week over Washington. They could be due for a letdown. And who better to serve it to them than old Eli, potentially on his way out of the division for good.

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Marcas Grant: Jimmy Garoppolo will be a top 10 quarterback nfl. Last week, Garoppolo looked poised and in command of the offense against a much better Bears defense, throwing for 293 yards (and an interception that could rightfully have been called a fumble) and moving the offense effectively.I needed this today.

Thank you for the awesome tips, illustrations, and instructions. Hi Travis, really well written article, with lots of thought and detail. I love your style in doing that 301 from your own site, classy. Finesse and attention to detail are sadly rare in the digital marketing world, so thanks for showcasing the little things which different some of us. Thanks for the easy to follow directions. It worked like a charm.

I posted this article to my business Facebook page. As a small business it is always nice to find ways to do things ourselves in order to cut our costs. This article was easy to follow and I am looking forward to our clients reviews.

My best guess is that results may depend on when the listing was created. When I created links for this tutorial, I created the link on an iPhone 5.

Thanks for putting this info together.

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Any idea why that would be happening. I noticed in your response to the previous comment you said that the link has to be generated from a mobile phone in order to work correctly. For reference, I am in the US, and your link above opens and goes straight to the star review on both my computer and iPhone. However, I have not been able to get it to work for our Google My Business Page. For reference I have 4 separate locations all tied together on Google My Business.

However, the link(s) will not open to the star review page ever, they only open to our Google page and would require a person to then select the leave a review, etc. When creating the link, I followed your instructions precisely, and tried creating the link to one of our locations Google pages specifically.

Thank you in advance for sharing the article and any ideas. Thanks for the note about changing the 3 to a 5 in the desktop URL to pre-fill a 5 star rating.

That was actually an issue I discovered when writing a previous version of this article.


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