Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0001 362200 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0639 1212848 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0639 1212848 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1347 4046176 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1360 4073504 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1367 4091216 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1367 4108224 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1374 4115960 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1374 4115960 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1808 4351648 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1822 4369048 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1828 4389976 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.1916 4452480 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.1916 4452736 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

## Vectors 1

A vector is the difference between two points. If two points areandthen we can write the difference(when written in a column between brackets) asThe arrow abovemeans that we are going from the pointto the pointIf we swapandthen we are going fromtoand writeThis is shown below.

We can also write points as pairs of coordinates and vectors as one number above another. If andthen

$\vec{B-A}=\begin{pmatrix}7\\10\end{pmatrix}-\begin{pmatrix}3\\2\end{pmatrix}=\begin{pmatrix}4\\8\end{pmatrix}$
.
Note that the top number when a vector is written as a column vector is the distance moved in the x direction and the bottom number is the distance moved in the y direction.

The magnitude of a vector is its length or the distance from end to end. We can find the magnitude of a vector written in column form using Pythagoras theorem. For A and B as above, to get from A to B we must go 4 in the x – direction and 8 in the y – direction. These are at right angles as shown below.

Then the length of

We can add and multiply vectors in the natural way so that ifthen and