<story><title>Look, ma, no matrices</title><url>https://enkimute.github.io/LookMaNoMatrices/</url></story><parent_chain><item><author>skhunted</author><text>Your comment is interesting to me.  A few days ago someone asked why math classes don’t the how and why and rather tend to just present the formulas for the operations and tell people to compute.  Here you are focusing on what the operations do rather than on why they work.  It’s an interesting contrast and one that teachers of mathematics have to balance.  The two questions,&lt;p&gt;How does it work?&lt;p&gt;Why does it work?&lt;p&gt;Can’t always both be answered well in a given course.</text></item><item><author>rhelz</author><text>Geometric Algebra was a complete mystery to me until I finally realized:  it is just polynomial multiplication, but with some quantities for which the order of multiplication matters, and which have a weird multiplication table:  i*i = 1, i*j = -j*i.  That&amp;#x27;s it.  Most intros present geometric product of two vectors:&lt;p&gt;(x1*i + y1*j) * (x2*i + y2*j)&lt;p&gt;as some deep mysterious thing, but its just the same FOIL polynomial multiplication you learned in freshman algebra:&lt;p&gt;(x1*i + y1*i)(x2*i+y2*j) = x1*x2*i*i + x1*y2*i*j + y1*x2*j*i + y1*y2*j*j&lt;p&gt;= (x1*x2 + y1*y2) + (x1*y2 - y2*x1)*i*j&lt;p&gt;The quantity in the first parenthesis, above, is the our old familiar dot product.  The quantity in the second parenthesis is our old friend the cross product, but expressed in a new dimension whose basis is i*j, and which--unlike the cross product--generalizes to any number of dimensions.   In GA its called the &amp;quot;wedge product&amp;quot;.&lt;p&gt;Once you &amp;quot;get&amp;quot; that, you find that doing things like deriving rotation formulas, etc, become easy, because you can apply all the skills you developed in algebra to solving geometric problems.</text></item></parent_chain><comment><author>aleph_minus_one</author><text>&amp;gt; A few days ago someone asked why math classes don’t the how and why and rather tend to just present the formulas for the operations and tell people to compute.&lt;p&gt;The lecturer typically do know quite well the how and why, but teaching these points takes a lot of time (you also have to explain a lot about the practical application until you are able to explain why this mathematical structure is helpful for the problem).&lt;p&gt;Since lecturers are typically very short of time in lectures, they teach the mathematics in a concise way and trust the students to be adults, capable of going to the library, and reading textbooks about the how and why by themselves if they are interested in this. At least in Germany, there is the clear mentality that if you are not capable of doing this, a university is a wrong place for you; you should better get vocational training instead.</text></comment>
<story><title>Look, ma, no matrices</title><url>https://enkimute.github.io/LookMaNoMatrices/</url></story><parent_chain><item><author>skhunted</author><text>Your comment is interesting to me.  A few days ago someone asked why math classes don’t the how and why and rather tend to just present the formulas for the operations and tell people to compute.  Here you are focusing on what the operations do rather than on why they work.  It’s an interesting contrast and one that teachers of mathematics have to balance.  The two questions,&lt;p&gt;How does it work?&lt;p&gt;Why does it work?&lt;p&gt;Can’t always both be answered well in a given course.</text></item><item><author>rhelz</author><text>Geometric Algebra was a complete mystery to me until I finally realized:  it is just polynomial multiplication, but with some quantities for which the order of multiplication matters, and which have a weird multiplication table:  i*i = 1, i*j = -j*i.  That&amp;#x27;s it.  Most intros present geometric product of two vectors:&lt;p&gt;(x1*i + y1*j) * (x2*i + y2*j)&lt;p&gt;as some deep mysterious thing, but its just the same FOIL polynomial multiplication you learned in freshman algebra:&lt;p&gt;(x1*i + y1*i)(x2*i+y2*j) = x1*x2*i*i + x1*y2*i*j + y1*x2*j*i + y1*y2*j*j&lt;p&gt;= (x1*x2 + y1*y2) + (x1*y2 - y2*x1)*i*j&lt;p&gt;The quantity in the first parenthesis, above, is the our old familiar dot product.  The quantity in the second parenthesis is our old friend the cross product, but expressed in a new dimension whose basis is i*j, and which--unlike the cross product--generalizes to any number of dimensions.   In GA its called the &amp;quot;wedge product&amp;quot;.&lt;p&gt;Once you &amp;quot;get&amp;quot; that, you find that doing things like deriving rotation formulas, etc, become easy, because you can apply all the skills you developed in algebra to solving geometric problems.</text></item></parent_chain><comment><author>rhelz</author><text>&amp;#x2F;&amp;#x2F; focusing on what the operations do rather than why &amp;#x2F;&amp;#x2F;&lt;p&gt;YMMV, of course, but in general, I always found it easier to understand the &lt;i&gt;why&lt;/i&gt; once I understood the &lt;i&gt;what&lt;/i&gt; and the &lt;i&gt;how&lt;/i&gt;, rather than the other way around.</text></comment>