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Finding the Magnitude of the Sum of Two Vectors Find the magnitude of the sum of a 15-km displacement and a 25-km displacement when the angle between them is 90 and when the angle between them is 135

C h a p t e r 1 3 : H o d g e p o d g e : A C h a p t e r o f M i s c e l l a n e o u s Te c h n i q u e s

Sketch the two displacement vectors, A and B, and the angle between them Known:

Unknown:

90 135 R

.

When the angle is 90 , use the Pythagorean theorem to find the magnitude of the resultant vector R 2 A2 B 2 R A2 B2 (25 km)2 (15 km)2 Substitute A 25 km, B 15 km 29 km When the angle does not equal 90 , use the law of cosines to find the magnitude of the resultant vector R 2 A2 B 2 2AB(cos 2) R A2 B2 2AB(cos (15 km)2

2(25 km)(15 km)(cos 135 )

15 km,

135

This program starts by initializing $reports, the array that will hold all the news reports Then $url is assigned the location of the Yahoo! News RSS feed for the search term in $search after $search has been converted to a form that can be passed in a URL using the rawurlencode() function The feed is then called up using the file_get_contents() function, preceded by an @ symbol to suppress any error messages The result is then placed in $xml If it is FALSE or an empty string, then a single-element array containing the value FALSE is returned Then, because the SimpleXML routines to be used later don t appear to work with CDATA (XML character data; see plug-in 76, Search Yahoo!, for details), the next few lines remove the CDATA tags and convert characters that might clash with SimpleXML into entities it understands Finally, $xml is converted into a SimpleXML object using the simplexml_load_string() function and the result is stored in $sxml Next, a foreach loop is used to iterate through all the elements of $sxml->channel>item, assigning each in turn to the object $item Inside the loop, the first thing that happens is the variable $flag is set to FALSE If it is later set to TRUE, then a title was found that was too similar to a previous one The variable $url is then extracted and the string $date is created from a timestamp by using the strtotime() and date() functions After that, the title and publishing site name are extracted into $title and $site, after exploding the title into the array $temp to split $site out of the title, where it was stored inside a pair of brackets The news story is then saved into the variable $desc To prevent similar stories being returned, a for loop is then used to iterate through all the saved stories in the $reports array Using the similar_text() function, each previous title is compared to the current one and, if it is more than 70 percent similar, the variable $flag is set to TRUE and a break command is issued to break out of the loop, as no further duplication checking is necessary At the tail end of the loop, as long as $flag doesn t have a value of TRUE and $desc actually contains some text, the story parts are grouped into an array that is then assigned to the next available element of $reports The plug-in returns a two-element array in which the first element is the number of news stories returned and the second is the $reports array.

To use VBA syntax inside MDX, you use VBA! followed by the name of the function Here we are looking at VBA Format For Excel functions you would type Excel! first

Are the units correct Each answer is a length measured in kilometers Do the signs make sense The sums are positive Are the magnitudes realistic The magnitudes are in the same range as the two combined vectors, but longer This is because each resultant is the side opposite an obtuse angle The second answer is larger than the first, which agrees with the graphical representation