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Mathematics - Mathematical Analysis Vectors, Lines and Planes by drifter1

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mathematics · @drifter1 · (edited)
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Mathematics - Mathematical Analysis Vectors, Lines and Planes
<html>
<p><img src="https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/lineplane/lineplane2.gif" width="250" height="184"/></p>
<p><a href="https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/lineplane/lineplane2.gif">Source</a></p>
<h2>Introduction</h2>
<p>Hello it's a me again drifter1!</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;It's <strong>Easter Holiday time</strong> for me and so I got back home to chill for 2 weeks with my family and friends. In the first week I had some time for posting, but during the weekend we went to my Aunt and so I don't had time to post! But here I am again to &nbsp;post for you Steemians!</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;Today's topic is<strong> Mathematical Analysis</strong> again and more specifically <strong>Vectors and equations for Lines and Planes</strong>! I already<strong> covered a lot about Vectors</strong> during my <strong>Physics </strong>"category" and so I suggest you to read the <strong>post </strong>about it <a href="https://steemit.com/physics/@drifter1/physics-vector-math-and-operations">here</a>. We will make a<strong> quick recap </strong>and then continue with some <em>more things</em> that I didn't had to talk about in Physics! After that we will get into Lines and Planes...</p>
<p>So, without further do, let's dive straight into it!</p>
<p><br></p>
<h2>Quick recap</h2>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Vectors are <strong>arrows that point from one point to another and describe the direction of movement</strong> and also by <strong>how much we move in each of the x, y or even z coordinates</strong> in 1-, 2- or 3-dimensional space (<strong>meter</strong>).</p>
<blockquote>From the Physics post (tweaked)...</blockquote>
<p>Two vectors are <strong>equal </strong>when they have the <strong>same direction and meter</strong>.</p>
<p>The meter |A| of vector A is of course equal to:</p>
<p><img src="http://quicklatex.com/cache3/db/ql_dd70d7783afed004c654cba1472faedb_l3.png"/></p>
<p>via <a href="http://quicklatex.com/cache3/db/ql_dd70d7783afed004c654cba1472faedb_l3.png">quicklatex</a>&nbsp;</p>
<p>where A = (P1P2) with P1 = (x1, y1, z1, ...) and P2 = (x2, y2, z2, ...)</p>
<p>Vectors can be analyzed into <strong>components </strong>for each axis.</p>
<p>We can of course do some <strong>operatios </strong>on vectors that include:</p>
<ul>
  <li><strong>Addition </strong>-&gt; A + B = C</li>
  <li><strong>Substraction </strong>-&gt; A - B = D</li>
  <li><strong>Scalar product </strong>-&gt; A*B = |A| |B| cosφ and so a real number</li>
  <li><strong>Cross product</strong> -&gt; AxB = ABsinφ and so a new vector [calculated using a determinant]</li>
</ul>
<p>&nbsp;&nbsp;&nbsp;&nbsp;To have a common language we use <strong>unit vectors</strong> for each axis of a linear space, where an unit vector has meter 1.</p>
<blockquote>To make a vector into a unit vector (meter 1) you simply divide it with it's meter!</blockquote>
<p><br></p>
<h2>Vector tangents and normals</h2>
<p><img src="https://ltcconline.net/greenl/courses/202/vectorFunctions/tannor11.gif" width="328" height="248"/></p>
<p><a href="https://ltcconline.net/greenl/courses/202/vectorFunctions/tannorm.htm">Source</a></p>
<ul>
  <li>A vector/line is tangent to an vector/line if they are <strong>parallel </strong>to each other.</li>
  <li>On the other hand it's a normal when they are <strong>vertically across</strong>.</li>
</ul>
<p>&nbsp;&nbsp;&nbsp;&nbsp;To find a parallel (tangent) vector at a specific point of a line we simply find a unit vector with the same angle λ as the line. On the other hand for finding a normal vector we find a vector with angle -1/λ, but there is also a small trick if we already found the answer of the tangent.</p>
<p><br></p>
<p><strong>Example</strong></p>
<p>For the line <img src="http://quicklatex.com/cache3/47/ql_0881497beaf6e81334ad3127e6e9f947_l3.png"/>&nbsp;&nbsp; (<a href="http://quicklatex.com/cache3/47/ql_0881497beaf6e81334ad3127e6e9f947_l3.png">quicklatex</a>) at x = y = 1 we have:</p>
<p><img src="http://quicklatex.com/cache3/13/ql_df21f46a650fbda9a9e64ebd3bd15113_l3.png"/>&nbsp; (<a href="http://quicklatex.com/cache3/13/ql_df21f46a650fbda9a9e64ebd3bd15113_l3.png">quicklatex</a>)</p>
<p>So, the vector we search for is:</p>
<p>v = 2i + 3j which has an angle of 3/2</p>
<p>The corresponding unit vector is:</p>
<p><img src="http://quicklatex.com/cache3/fd/ql_ce68946453707af8dcd5271e6bd8befd_l3.png"/>&nbsp;(<a href="http://quicklatex.com/cache3/fd/ql_ce68946453707af8dcd5271e6bd8befd_l3.png">quicklatex</a>)</p>
<p>This is the tangent at (1, 1).</p>
<p>We could also use the negatve vector -u.</p>
<p>For the normal vector we simply find the negative inverse of u.</p>
<p>To do that we swap the places of the components and change the sign of one of them.</p>
<p>So, a normal vector is:</p>
<p><img src="http://quicklatex.com/cache3/1b/ql_12aa85ba6f67908b9109b1d448ba531b_l3.png"/>&nbsp;(<a href="http://quicklatex.com/cache3/1b/ql_12aa85ba6f67908b9109b1d448ba531b_l3.png">quicklatex</a>)</p>
<p><br></p>
<p>Two more <strong>useful things </strong>are the following:</p>
<ol>
  <li>Two vectors are across if the scalar product of them is zero (u.v = 0).</li>
  <li>Two vectors are parallel if the cross product of them is zero (uxv = 0)</li>
</ol>
<p><br></p>
<h2>Vector projection</h2>
<p><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/9/98/Projection_and_rejection.png/200px-Projection_and_rejection.png" width="200" height="180"/></p>
<p><a href="https://upload.wikimedia.org/wikipedia/commons/thumb/9/98/Projection_and_rejection.png/200px-Projection_and_rejection.png">Source</a></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;The vector projection of a vector u onto a line parallel to another non-zero vector v is the orthogonal projection of u straight into the line parallel to v.</p>
<p>From <a href="https://en.wikipedia.org/wiki/Vector_projection">wikipedia</a></p>
<p>It can be noted as proj (u, v) which means projection of u onto v.</p>
<p>To calculate it we use:</p>
<p><em><strong>proj (u, v) = |u|cosθ = u * v/|v|</strong></em></p>
<p>where u*v is the scalar or dot product and θ the angle between u and v.</p>
<p><br></p>
<h2>Cartesian coordination space</h2>
<p>The <strong>Cartesian space</strong> is defined using 3 axes (x, y and z).</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;Each point P is represented using values given to each of those coordinates that represent the position on each axis.</p>
<p>The center of the space is O(0, 0, 0).</p>
<p>In this space we can <strong>define plane</strong>s like:</p>
<ul>
  <li>xy where each point has z = 0</li>
  <li>xz where each point has y = 0</li>
  <li>yz where each point has x = 0</li>
</ul>
<p>Using even more <strong>restrictions </strong>we can define more an more <strong>subspaces </strong>if this Cartesian space.</p>
<p>As already shown before vectors are represented using:</p>
<p><em><strong>u = OP = xi + yj + zk</strong></em></p>
<p>where (x, y, z) is the point P and i, j, k are the unit vector on each axis.</p>
<p>The <strong>meter </strong>is of course equal to the square root of the squares on of each coordinate.</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;Using the same concept we can calculate the distance between two points P1 = (x1, y1, z1) and P2 = (x2, y2, z2) in space, as the meter of the vector P1P2.</p>
<p>The equation of <strong>distance </strong>is:</p>
<p><img src="http://quicklatex.com/cache3/cd/ql_be63da979e4bef0b9b62f58cac97d2cd_l3.png"/>&nbsp;(<a href="http://quicklatex.com/cache3/cd/ql_be63da979e4bef0b9b62f58cac97d2cd_l3.png">quicklatex</a>)</p>
<p><br></p>
<h2>Lines in space</h2>
<p>A line in space can be <strong>represented </strong>by only two things:</p>
<ul>
  <li>A point of the line (P)</li>
  <li>The slope/angle λ</li>
</ul>
<p>A straight line is simply y = λx.</p>
<p><br></p>
<p>For more complex ones we use two <strong>representations</strong>:</p>
<ol>
  <li>Vector representation</li>
  <li>Parametric representation</li>
</ol>
<h3><br></h3>
<h3>Vector representation</h3>
<p>A line L parallel to a vector v that surpasses the point P0 = (x0, y0, z0) is:</p>
<p><em><strong>r(t) = r0 + i*v</strong></em></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;where r is the position vector of the point P(x, y, z) of L and r0 is the position vector of the point P0.</p>
<p><br></p>
<h3>Parametric representation</h3>
<p>By taking the components of the previous one we end up with the parametric representation.</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;For a line that contains the point P0 = (x0, y0, z0) and is parallel to the vector v = u1i + u2j + u3k we have:</p>
<p><em><strong>x = x0 + t*u1</strong></em></p>
<p><em><strong>y = y0 + t*u2</strong></em></p>
<p><em><strong>z = &nbsp;z0 + t*u3</strong></em></p>
<p>where t is any real number.</p>
<p><br></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;So, to find a line we simply need to find a parallel vector to that line and need to know a point where the line passes by, cause the vector and line have the same slope/angle (tangent vector).</p>
<p><br></p>
<h2>Planes in space</h2>
<p>Using the same concept we can also define a plane.</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;A plane that contains the point P0(x0, y0, z0) and is vertically across to a vector n = Ai +Bj + Ck (normal) is of the form:</p>
<p><em><strong>n*(P0P) = 0</strong></em></p>
<p>or</p>
<p><em><strong>A(x - x0) + B(y - y0) + C(z - z0) = 0</strong></em></p>
<p>or</p>
<p><em><strong>Ax + By + Cz = D </strong></em>where<em><strong> D = Ax0 + By0 + Cz0</strong></em></p>
<p><br></p>
<p>So, to find a plane we just have to find a normal vector to that plane and have a point given.</p>
<p><br></p>
<p>Because the Plane types are more advanced I leave them for a post on their own...</p>
<p><br></p>
<p>And this is actually it and I hope that you enjoyed it!</p>
<p>Bye!</p>
</html>
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@alexdory ·
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Very nice project, made me remember my math years in college. 
It's very strange for me (non native English) to try to understand an English math demonstration :D 
I appreciate your general public approach, I am still dreaming of complementary education done on Steemit :D
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@minnowsupport ·
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<p>Congratulations!  This post has been upvoted from the communal account, @minnowsupport, by drifteritis from the Minnow Support Project. It's a witness project run by aggroed, ausbitbank, teamsteem, theprophet0, someguy123, neoxian, followbtcnews, and netuoso. The goal is to help Steemit grow by supporting Minnows.  Please find us at the <a href="https://discord.gg/HYj4yvw"> Peace, Abundance, and Liberty Network (PALnet) Discord Channel</a>.  It's a completely public and open space to all members of the Steemit community who voluntarily choose to be there.</p> <p>If you would like to delegate to the Minnow Support Project you can do so by clicking on the following links: <a href="https://v2.steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=minnowsupport&amp;vesting_shares=102530.639667%20VESTS">50SP</a>, <a href="https://v2.steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=minnowsupport&amp;vesting_shares=205303.639667%20VESTS">100SP</a>, <a href="https://v2.steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=minnowsupport&amp;vesting_shares=514303.639667%20VESTS">250SP</a>, <a href="https://v2.steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=minnowsupport&amp;vesting_shares=1025303.639667%20VESTS">500SP</a>, <a href="https://v2.steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=minnowsupport&amp;vesting_shares=2053030.639667%20VESTS">1000SP</a>, <a href="https://v2.steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=minnowsupport&amp;vesting_shares=10253030.639667%20VESTS">5000SP</a>. <br><strong>Be sure to leave at least 50SP undelegated on your account.</strong></p>
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@teslalifestyle · (edited)
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Just checking in after a hiatus, glad to see you are still at it. Was checking out the differential equations articles and found them really useful. Is this going to be a part of a Vector Calculus type series or still keeping it pretty general?
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@drifter1 ·
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I try to write things in an easy way that can be understood by anyone, even those that dont know a lot of maths...
Mathematical Analysis is a series of its own and the posts that follow will be much more advanced then the first parts of this series that contained only limits, derivatives, integrals and series...
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@teslalifestyle ·
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very cool looking forward seeing how the series develops
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