{"id":420,"date":"2012-09-29T18:17:55","date_gmt":"2012-09-30T01:17:55","guid":{"rendered":"http:\/\/zackmdavis.net\/blog\/?p=420"},"modified":"2012-09-29T18:17:55","modified_gmt":"2012-09-30T01:17:55","slug":"blades","status":"publish","type":"post","link":"http:\/\/zackmdavis.net\/blog\/2012\/09\/blades\/","title":{"rendered":"Blades"},"content":{"rendered":"

What is a vector<\/em> in Euclidean space? Some might say it's an entity characterized by possessing a magnitude<\/em> and a direction<\/em>. But scholars of the geometric algebra (such as Eric Chisolm<\/a> and Dorst et al.<\/em><\/a>) tell us that it's better to decompose the idea of direction<\/em> into the two ideas of subspace attitude<\/em> (our vector's quality of living in a particular line) and orientation<\/em> (its quality of pointing in a particular direction in that line, and not the other). On this view, a vector is an attitudinal oriented length element<\/em>. But having done this, it becomes inevitable that we should want to talk about attitudinal oriented area<\/em> (volume, 4-hypervolume, &c.<\/em>) elements. To this end we introduce the outer<\/em> or wedge product \u2227 on vectors. It is bilinear<\/em>, it is anticommutative<\/em> (swapping the order of arguments swaps the sign, so a<\/strong>\u2227b<\/strong> = \u2013b<\/strong>\u2227a<\/strong>), and that's all you need to know.<\/p>\n

Suppose we have two vectors a<\/strong> and b<\/strong> in Euclidean space and also a basis for the subspace that the vectors live in, e<\/strong>1<\/sub> and e<\/strong>2<\/sub>, so that we can write a<\/strong> := a1<\/sub>e<\/strong>1<\/sub> + a2<\/sub>e<\/strong>2<\/sub> and b<\/strong> := b1<\/sub>e<\/strong>1<\/sub> + b2<\/sub>e<\/strong>2<\/sub>. Then the claim is that the outer product a<\/strong>\u2227b<\/strong> can be said to represent a generalized vector (call it a 2-blade<\/em>\u2014and in general, when we wedge k<\/em> vectors together, it's a k<\/em>-blade) with a subspace attitude of the plane that our vectors live in and a magnitude equal to the area of the parallelogram spanned by them. Following Dorst et al<\/em>., let's see what happens when we expand a<\/strong>\u2227b<\/strong> in terms of our basis\u2014<\/p>\n

<\/p>\n

a<\/strong>\u2227b<\/strong> = (a1<\/sub>e<\/strong>1<\/sub> + a2<\/sub>e<\/strong>2<\/sub>)\u2227(b1<\/sub>e<\/strong>1<\/sub> + b2<\/sub>e<\/strong>2<\/sub>)<\/p>\n

= a1<\/sub>e<\/strong>1<\/sub>\u2227(b1<\/sub>e<\/strong>1<\/sub> + b2<\/sub>e<\/strong>2<\/sub>) + a2<\/sub>e<\/strong>2<\/sub>\u2227(b1<\/sub>e<\/strong>1<\/sub> + b2<\/sub>e<\/strong>2<\/sub>)<\/p>\n

= a1<\/sub>e<\/strong>1<\/sub>\u2227b1<\/sub>e<\/strong>1<\/sub> + a1<\/sub>e<\/strong>1<\/sub>\u2227b2<\/sub>e<\/strong>2<\/sub> + a2<\/sub>e<\/strong>2<\/sub>\u2227b1<\/sub>e<\/strong>1<\/sub> + a2<\/sub>e<\/strong>2<\/sub>\u2227b2<\/sub>e<\/strong>2<\/sub><\/p>\n

But the anticommutativity property implies that the outer product of a vector with itself is zero<\/em>, because e<\/strong>\u2227e<\/strong> = \u2013e<\/strong>\u2227e<\/strong>. So we have<\/p>\n

(a1<\/sub>b2<\/sub> \u2013 a2<\/sub>b1<\/sub>)e<\/strong>1<\/sub>\u2227e<\/strong>2<\/sub><\/p>\n

It's a determinant! And since determinants tell us about the oriented volumes of parallelepipeds, we can see why these blades defined by this outer product are a sensible generalization of the vector<\/em> idea. And none can doubt that they shall play but ever such an essential role in our vaunted geometric algebra!<\/p>\n","protected":false},"excerpt":{"rendered":"

What is a vector in Euclidean space? Some might say it's an entity characterized by possessing a magnitude and a direction. But scholars of the geometric algebra (such as Eric Chisolm and Dorst et al.) tell us that it's better … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[7],"tags":[43],"_links":{"self":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/420"}],"collection":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/comments?post=420"}],"version-history":[{"count":5,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/420\/revisions"}],"predecessor-version":[{"id":425,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/420\/revisions\/425"}],"wp:attachment":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/media?parent=420"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/categories?post=420"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/tags?post=420"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}