vectors的十字架 - 概念

决定簇的一个非常重要的应用是它可用于定义载体的十字乘产物，因此这将我们带回载体。让我们回顾一个想法。矢量A组件是A，B，C的3D矢量V可以根据单位向量I，J和K写入，其中I是正X方向上的单位向量，J是正y方向的单位向量k是正Z方向的单位向量，所以它只是像XYZ，IJK一样的字母顺序，这就是你将写载体v你将它写为AAH AAH A Times i Plus B次J加权，组件仍然很容易看到。
现在让我们假设我们在这个形式中有两个向量，j，k形式然后是横向产品被定义的v十字架，等于决定因素和第一行，我把3单位向量i，j和k旋转，j和k然后i跨越第二行横向产品v中的第一载体的组件，以及在最后一行上，我将横向产品中的第二个向量的组件放在横向产品w中，因此您将这两个右图的顺序非常重要？第一个在横向产品中的第一个，你把第二排放在第三行中的第二行。让我们在一个例子中尝试一下。
So I have two vectors v equals 4, -5, -3, w equals 2, 2, 1 I first have to put these in i j k form so that's really easy because all you have to do is just fill in the i-5j-3k same for this, 2i plus 2j plus 1k it's just k and then we cross w is the determinant i, j, k so you always put i, j, k across the top and then 4, -5, -3 those are the components of v and then the components of w 2, 2 and 1 and you expand this the way you would a normal determinant so I'll expand across the top row I have i times this minor -5, -3, 2, 1 then I have to subtract j times it's minor 4, -3, 2, 1 then I add k times its minor oops 4, -5, 2, 2 okay and then I just compute these two by two determinants so this one is going to be -5--6 so -5+6 which is 1 that gives me i times 1 minus j this will be 4--6, 4+6 10 so j times 10 plus k I have 8--10 8+10 is 18 so the answer is i-10j+18k and of course you can put that into regular component form if you like, v cross w is 1,-10, 18 so depending on what what form your teacher wants either of these is an acceptable answer.