交叉产品据/h1>
已经有一个帐户?据一种href="//www.parkandroid.com/account/login/?next=/wiki/cross-product-definition/" class="ax-click" data-ax-id="clicked_signup_modal_login" data-ax-type="link">这里登录。据/a>
相关......据/h4>
- 几何学据/span>>据/span>
这据strong>交叉产品据/strong>是A.据一种href="//www.parkandroid.com/wiki/vector-introduction/" class="wiki_link" title="向量" target="_blank">向量据/a>操作在三维的载体上起作用的操作,并导致三维的另一个载体。与此相反据一种href="//www.parkandroid.com/wiki/dot-product-definition/" class="wiki_link" title="点产品" target="_blank">点产品据/a>,可以在2-D和3-D空间中定义,交叉产品仅在3D空间中定义。另一个不同之处在于,虽然点 - 产品输出标量数,但横向产品输出另一个向量。据/p>
定义据/h2>
交叉产品的代数解释被定义为据/p>
如果据span class="katex"> 一种据/span> =据/span>(据/span>一种据/span>1据/span>那据/span>一种据/span>2据/span>那据/span>一种据/span>3.据/span>)据/span>和据span class="katex"> B.据/span> =据/span>(据/span>B.据/span>1据/span>那据/span>B.据/span>2据/span>那据/span>B.据/span>3.据/span>)据/span>是3 d空间中的2个矢量,然后是据strong>交叉产品据/strong> 一种据/span> ×据/span>B.据/span> 可以通过以下方式定义:据/p>
- 作为决定因素据span class="katex-display"> |据/span>|据/span>|据/span>|据/span>|据/span>|据/span>一世据/span>一种据/span>1据/span>B.据/span>1据/span>j据/span>一种据/span>2据/span>B.据/span>2据/span>K.据/span>一种据/span>3.据/span>B.据/span>3.据/span>|据/span>|据/span>|据/span>|据/span>|据/span>|据/span>那据/span>使用第一行;据/li>
- 作为据span class="katex"> (据/span>一种据/span>2据/span>B.据/span>3.据/span>-据/span>一种据/span>3.据/span>B.据/span>2据/span>那据/span>一种据/span>3.据/span>B.据/span>1据/span>-据/span>一种据/span>1据/span>B.据/span>3.据/span>那据/span>那据/span>一种据/span>1据/span>B.据/span>2据/span>-据/span>一种据/span>2据/span>B.据/span>1据/span>)据/span>在矢量表示法。据!-- end-definition -->
找到杂交产品据span class="katex"> V.据/span> =据/span>(据/span>3.据/span>那据/span>6.据/span>那据/span>8.据/span>)据/span>和据span class="katex"> W.据/span> =据/span>(据/span>2据/span>那据/span>-据/span>4.据/span>那据/span>7.据/span>)据/span>。据/span>
方法1:据/strong>
作为决定因素,我们有据/p>|据/span>|据/span>|据/span>|据/span>|据/span>|据/span>一世据/span>3.据/span>2据/span>j据/span>6.据/span>-据/span>4.据/span>K.据/span>8.据/span>7.据/span>|据/span>|据/span>|据/span>|据/span>|据/span>|据/span>=据/span>=据/span>=据/span>(据/span>6.据/span>(据/span>7.据/span>)据/span>-据/span>8.据/span>(据/span>-据/span>4.据/span>)据/span>)据/span>一世据/span>-据/span>(据/span>3.据/span>(据/span>7.据/span>)据/span>-据/span>2据/span>(据/span>8.据/span>)据/span>)据/span>j据/span>+据/span>(据/span>3.据/span>(据/span>-据/span>4.据/span>)据/span>-据/span>(据/span>6.据/span>)据/span>(据/span>2据/span>)据/span>)据/span>K.据/span>7.据/span>4.据/span>一世据/span>-据/span>5.据/span>j据/span>-据/span>2据/span>4.据/span>K.据/span>(据/span>7.据/span>4.据/span>那据/span>-据/span>5.据/span>那据/span>-据/span>2据/span>4.据/span>)据/span>。据/span>
方法2:据/strong>
简单地使用定义(b)的公式,我们有据/p>(据/span>V.据/span>2据/span>W.据/span>3.据/span>-据/span>V.据/span>3.据/span>W.据/span>2据/span>那据/span>V.据/span>3.据/span>W.据/span>1据/span>-据/span>V.据/span>1据/span>W.据/span>3.据/span>那据/span>那据/span>V.据/span>1据/span>W.据/span>2据/span>-据/span>V.据/span>2据/span>W.据/span>1据/span>)据/span>=据/span>(据/span>6.据/span>(据/span>7.据/span>)据/span>-据/span>8.据/span>(据/span>-据/span>4.据/span>)据/span>那据/span>2据/span>(据/span>8.据/span>)据/span>-据/span>3.据/span>(据/span>7.据/span>)据/span>那据/span>3.据/span>(据/span>-据/span>4.据/span>)据/span>-据/span>(据/span>6.据/span>)据/span>(据/span>2据/span>)据/span>)据/span>=据/span>(据/span>7.据/span>4.据/span>那据/span>-据/span>5.据/span>那据/span>-据/span>2据/span>4.据/span>)据/span>。据/span>□据/span>
找到载体的横向产品据span class="katex"> V.据/span> =据/span>3.据/span>一世据/span>-据/span>2据/span>j据/span>-据/span>K.据/span>和据span class="katex"> W.据/span> =据/span>4.据/span>一世据/span>+据/span>3.据/span>j据/span>+据/span>2据/span>K.据/span>。据/span>
我们可以将横向产品作为以下矩阵的决定因素获取:据/p>
|据/span>|据/span>|据/span>|据/span>|据/span>|据/span>一世据/span>3.据/span>4.据/span>j据/span>-据/span>2据/span>3.据/span>K.据/span>-据/span>1据/span>2据/span>|据/span>|据/span>|据/span>|据/span>|据/span>|据/span>=据/span>(据/span>-据/span>2据/span>(据/span>2据/span>)据/span>-据/span>3.据/span>(据/span>-据/span>1据/span>)据/span>)据/span>一世据/span>-据/span>(据/span>3.据/span>(据/span>2据/span>)据/span>-据/span>4.据/span>(据/span>-据/span>1据/span>)据/span>)据/span>j据/span>+据/span>(据/span>3.据/span>(据/span>3.据/span>)据/span>-据/span>4.据/span>(据/span>-据/span>2据/span>)据/span>)据/span>K.据/span>=据/span>-据/span>一世据/span>-据/span>1据/span>0.据/span>j据/span>+据/span>1据/span>7.据/span>K.据/span>。据/span>□据/span>
特性据/h2>
交叉产品的一些主要属性如下:据/p>
1.杂交产品不是换向。那是,据span class="katex"> 一种据/span> ×据/span>B.据/span> 据/span>=据/span>B.据/span> ×据/span>一种据/span> 。通过右侧螺丝规则,据span class="katex"> 一种据/span> ×据/span>B.据/span> 和据span class="katex"> B.据/span> ×据/span>一种据/span> 有相反的方向,所以据span class="katex"> 一种据/span> ×据/span>B.据/span> =据/span>-据/span>B.据/span> ×据/span>一种据/span> 。2.交叉产品是关于载体添加的分配:据span class="katex"> 一种据/span> ×据/span>(据/span>B.据/span> +据/span>C据/span> )据/span>=据/span>一种据/span> ×据/span>B.据/span> +据/span>一种据/span> ×据/span>C据/span> 。3.据span class="katex"> 0.据/span>×据/span>一种据/span> =据/span>一种据/span> ×据/span>0.据/span>=据/span>0.据/span>。4.据span class="katex"> C据/span>(据/span>一种据/span> ×据/span>B.据/span> )据/span>=据/span>(据/span>C据/span>一种据/span> )据/span>×据/span>B.据/span> =据/span>一种据/span> ×据/span>(据/span>C据/span>B.据/span> )据/span>。5。据span class="katex"> 一种据/span> ×据/span>一种据/span> =据/span>0.据/span>。6.在反射(即镜像)下,矢量变化标志的所有组件。因此,对于据span class="katex"> 一种据/span> ×据/span>B.据/span> 那据/span>我们有据span class="katex"> (据/span>-据/span>一种据/span> )据/span>×据/span>(据/span>-据/span>B.据/span> )据/span>=据/span>一种据/span> ×据/span>B.据/span> 。所以据span class="katex"> 一种据/span> ×据/span>B.据/span> 在反思下保持不变。据/p>
所有这些属性都可以从横向产品的定义导出,并留下作为读者证明的练习。据/p>
给予两个向量据span class="katex"> V.据/span> =据/span>3.据/span>一世据/span>-据/span>2据/span>j据/span>-据/span>K.据/span>和据span class="katex"> W.据/span> =据/span>4.据/span>一世据/span>+据/span>3.据/span>j据/span>+据/span>2据/span>K.据/span>, 找据/p>
一种)据span class="katex"> V.据/span> ×据/span>W.据/span>
b)据span class="katex"> W.据/span> ×据/span>V.据/span> 。据/p>
一种)据span class="katex"> V.据/span> ×据/span>W.据/span> :据/span>
我们可以通过计算决定蛋白来获得横向产品:据span class="katex-display"> |据/span>|据/span>|据/span>|据/span>|据/span>|据/span>一世据/span>3.据/span>4.据/span>j据/span>-据/span>2据/span>3.据/span>K.据/span>-据/span>1据/span>2据/span>|据/span>|据/span>|据/span>|据/span>|据/span>|据/span>=据/span>(据/span>-据/span>2据/span>(据/span>2据/span>)据/span>-据/span>3.据/span>(据/span>-据/span>1据/span>)据/span>)据/span>一世据/span>-据/span>(据/span>3.据/span>(据/span>2据/span>)据/span>-据/span>4.据/span>(据/span>-据/span>1据/span>)据/span>)据/span>j据/span>+据/span>(据/span>3.据/span>(据/span>3.据/span>)据/span>-据/span>4.据/span>(据/span>-据/span>2据/span>)据/span>)据/span>K.据/span>=据/span>-据/span>一世据/span>-据/span>1据/span>0.据/span>j据/span>+据/span>1据/span>7.据/span>K.据/span>。据/span>b)据span class="katex"> W.据/span> ×据/span>V.据/span> :据/span>
我们可以遵循相同的步骤,但快捷方式将在上面使用定理:据span class="katex-display"> W.据/span> ×据/span>V.据/span> =据/span>-据/span>(据/span>V.据/span> ×据/span>W.据/span> )据/span>=据/span>-据/span>(据/span>-据/span>一世据/span>-据/span>1据/span>0.据/span>j据/span>+据/span>1据/span>7.据/span>K.据/span>)据/span>=据/span>一世据/span>+据/span>1据/span>0.据/span>j据/span>-据/span>1据/span>7.据/span>K.据/span>。据/span>□据/span>
还有一些相关的横向产品和DOT产品的属性:据/p>
一种)据span class="katex"> 一种据/span> ⋅据/span>(据/span>一种据/span> ×据/span>B.据/span> )据/span>=据/span>0.据/span>
b)据span class="katex"> B.据/span> ⋅据/span>(据/span>一种据/span> ×据/span>B.据/span> )据/span>=据/span>0.据/span>
C)据span class="katex"> 一种据/span> ×据/span>(据/span>B.据/span> ×据/span>C据/span> )据/span>=据/span>(据/span>一种据/span> ⋅据/span>C据/span> )据/span>B.据/span> -据/span>(据/span>一种据/span> ⋅据/span>B.据/span> )据/span>C据/span>
d)据span class="katex"> (据/span>一种据/span> ×据/span>B.据/span> )据/span>×据/span>C据/span> =据/span>(据/span>一种据/span> ⋅据/span>C据/span> )据/span>B.据/span> -据/span>(据/span>B.据/span> ⋅据/span>C据/span> )据/span>一种据/span>
e)据span class="katex"> ∥据/span>∥据/span>∥据/span>一种据/span> ×据/span>B.据/span> ∥据/span>∥据/span>∥据/span>2据/span>=据/span>∥据/span>一种据/span> ∥据/span>2据/span>∥据/span>∥据/span>∥据/span>B.据/span> ∥据/span>∥据/span>∥据/span>2据/span>-据/span>(据/span>一种据/span> ⋅据/span>B.据/span> )据/span>2据/span>(作为。。而被知道据一种href="//www.parkandroid.com/wiki/cross-product-lagranges-identity/?wiki_title=Lagrange's identity" class="wiki_link new" title="拉格朗日的身份" target="_blank" rel="nofollow">拉格朗日的身份据/a>)据/p>
如果我们意识到横向输出垂直于两个向量的向量,并且垂直载体的点乘积为零,则前两个属性易于理解。除了点和交叉产品的性质,其他人就像前两个一样,并且留给读者证明。据/p>
鉴于载体据span class="katex"> 一种据/span> =据/span>(据/span>-据/span>1据/span>那据/span>2据/span>那据/span>2据/span>)据/span>那据/span>B.据/span> =据/span>(据/span>0.据/span>那据/span>3.据/span>那据/span>4.据/span>)据/span>那据/span>C据/span> =据/span>(据/span>1据/span>那据/span>-据/span>2据/span>那据/span>0.据/span>)据/span>那据/span>表明它们确实满足上述四个属性a)通过d)。据/p>
一种)据/strong> 一种据/span> ⋅据/span>(据/span>一种据/span> ×据/span>B.据/span> )据/span>:据/span>
我们有据span class="katex-display"> 一种据/span> ×据/span>B.据/span> =据/span>⎣据/span>⎡据/span>一世据/span>-据/span>1据/span>0.据/span>j据/span>2据/span>3.据/span>K.据/span>2据/span>4.据/span>⎦据/span>⎤据/span>=据/span>(据/span>2据/span>那据/span>4.据/span>那据/span>-据/span>3.据/span>)据/span>⟹据/span>一种据/span> ⋅据/span>(据/span>一种据/span> ×据/span>B.据/span> )据/span>=据/span>(据/span>-据/span>1据/span>)据/span>(据/span>2据/span>)据/span>+据/span>2据/span>(据/span>4.据/span>)据/span>+据/span>2据/span>(据/span>-据/span>3.据/span>)据/span>=据/span>0.据/span>。据/span>b)据/strong> B.据/span> ⋅据/span>(据/span>一种据/span> ×据/span>B.据/span> )据/span>:据/span>
我们有据span class="katex-display"> 一种据/span> ×据/span>B.据/span> =据/span>⎣据/span>⎡据/span>一世据/span>-据/span>1据/span>0.据/span>j据/span>2据/span>3.据/span>K.据/span>2据/span>4.据/span>⎦据/span>⎤据/span>=据/span>(据/span>2据/span>那据/span>4.据/span>那据/span>-据/span>3.据/span>)据/span>⟹据/span>B.据/span> ⋅据/span>(据/span>一种据/span> ×据/span>B.据/span> )据/span>=据/span>0.据/span>(据/span>2据/span>)据/span>+据/span>3.据/span>(据/span>4.据/span>)据/span>+据/span>4.据/span>(据/span>-据/span>3.据/span>)据/span>=据/span>0.据/span>。据/span>C)据/strong> 一种据/span> ×据/span>(据/span>B.据/span> ×据/span>C据/span> )据/span>=据/span>(据/span>一种据/span> ⋅据/span>C据/span> )据/span>B.据/span> -据/span>(据/span>一种据/span> ⋅据/span>B.据/span> )据/span>C据/span> :据/span>
我们有据span class="katex"> 一种据/span> ×据/span>(据/span>B.据/span> ×据/span>C据/span> )据/span>=据/span>(据/span>-据/span>1据/span>4.据/span>那据/span>1据/span>3.据/span>那据/span>-据/span>2据/span>0.据/span>)据/span>,将为读者留下来展示如何。然后据span class="katex-display"> 一种据/span> ⋅据/span>C据/span> 一种据/span> ⋅据/span>B.据/span> 一种据/span> ×据/span>(据/span>B.据/span> ×据/span>C据/span> )据/span>⇒据/span>(据/span>-据/span>1据/span>4.据/span>那据/span>1据/span>3.据/span>那据/span>-据/span>2据/span>0.据/span>)据/span>=据/span>-据/span>1据/span>(据/span>1据/span>)据/span>+据/span>2据/span>(据/span>-据/span>2据/span>)据/span>+据/span>2据/span>(据/span>0.据/span>)据/span>=据/span>-据/span>5.据/span>=据/span>-据/span>1据/span>(据/span>0.据/span>)据/span>+据/span>2据/span>(据/span>3.据/span>)据/span>+据/span>2据/span>(据/span>4.据/span>)据/span>=据/span>1据/span>4.据/span>=据/span>(据/span>一种据/span> ⋅据/span>C据/span> )据/span>B.据/span> -据/span>(据/span>一种据/span> ⋅据/span>B.据/span> )据/span>C据/span> =据/span>-据/span>5.据/span>(据/span>0.据/span>那据/span>3.据/span>那据/span>4.据/span>)据/span>-据/span>1据/span>4.据/span>(据/span>1据/span>那据/span>-据/span>2据/span>那据/span>0.据/span>)据/span>=据/span>(据/span>0.据/span>那据/span>1据/span>5.据/span>那据/span>-据/span>2据/span>0.据/span>)据/span>-据/span>(据/span>1据/span>4.据/span>那据/span>-据/span>2据/span>8.据/span>那据/span>0.据/span>)据/span>=据/span>(据/span>-据/span>1据/span>4.据/span>那据/span>1据/span>3.据/span>那据/span>-据/span>2据/span>0.据/span>)据/span>。据/span>d)据/strong> (据/span>一种据/span> ×据/span>B.据/span> )据/span>×据/span>C据/span> =据/span>(据/span>一种据/span> ⋅据/span>C据/span> )据/span>B.据/span> -据/span>(据/span>B.据/span> ⋅据/span>C据/span> )据/span>一种据/span> :据/span>
我们有据span class="katex-display"> 一种据/span> ⋅据/span>C据/span> B.据/span> ⋅据/span>C据/span> (据/span>一种据/span> ×据/span>B.据/span> )据/span>×据/span>C据/span> ⇒据/span>(据/span>-据/span>6.据/span>那据/span>-据/span>3.据/span>那据/span>-据/span>8.据/span>)据/span>=据/span>-据/span>1据/span>(据/span>1据/span>)据/span>+据/span>2据/span>(据/span>-据/span>2据/span>)据/span>+据/span>2据/span>(据/span>0.据/span>)据/span>=据/span>-据/span>5.据/span>=据/span>0.据/span>(据/span>1据/span>)据/span>+据/span>3.据/span>(据/span>-据/span>2据/span>)据/span>+据/span>4.据/span>(据/span>0.据/span>)据/span>=据/span>-据/span>6.据/span>=据/span>(据/span>一种据/span> ⋅据/span>C据/span> )据/span>B.据/span> -据/span>(据/span>B.据/span> ⋅据/span>C据/span> )据/span>一种据/span> =据/span>-据/span>5.据/span>(据/span>0.据/span>那据/span>3.据/span>那据/span>4.据/span>)据/span>-据/span>(据/span>-据/span>6.据/span>)据/span>(据/span>-据/span>1据/span>那据/span>2据/span>那据/span>2据/span>)据/span>=据/span>(据/span>0.据/span>那据/span>-据/span>1据/span>5.据/span>那据/span>2据/span>0.据/span>)据/span>-据/span>(据/span>6.据/span>那据/span>-据/span>1据/span>2据/span>那据/span>-据/span>1据/span>2据/span>)据/span>=据/span>(据/span>-据/span>6.据/span>那据/span>-据/span>3.据/span>那据/span>-据/span>8.据/span>)据/span>。据/span>□据/span>
对于载体据/p>
一种据/span>=据/span>(据/span>1据/span>那据/span>0.据/span>那据/span>-据/span>1据/span>)据/span>那据/span>B.据/span>=据/span>(据/span>0.据/span>那据/span>2据/span>那据/span>1据/span>)据/span>那据/span>
什么是横向产品据span class="katex"> 一种据/span>×据/span>B.据/span>还是据/span>
几何定义据/h2>
考虑三维载体据span class="katex"> 一种据/span> 和据span class="katex"> B.据/span> , 然后让据span class="katex"> θ.据/span>是它们之间的角度。这据strong>交叉产品的几何解释据/strong>是垂直于两者的载体据span class="katex"> 一种据/span> 和据span class="katex"> B.据/span> (使用右侧规则)并具有定义为据/p>
∥据/span>一种据/span>×据/span>B.据/span>∥据/span>=据/span>∥据/span>一种据/span> ∥据/span>∥据/span>∥据/span>∥据/span>B.据/span> ∥据/span>∥据/span>∥据/span>罪据/span>θ.据/span>。据/span>
给予两个向量据span class="katex"> V.据/span> =据/span>3.据/span>一世据/span>+据/span>2据/span>j据/span>-据/span>0.据/span>K.据/span>和据span class="katex"> W.据/span> =据/span>5.据/span>一世据/span>-据/span>j据/span>+据/span>0.据/span>K.据/span>那据/span>如果两个向量之间的角度是的,则交叉产品的大小是多少据span class="katex"> 4.据/span>5.据/span>∘据/span>还是据/span>
我们有据/p>
∥据/span>V.据/span> ∥据/span>∥据/span>W.据/span> ∥据/span>=据/span>3.据/span>2据/span>+据/span>2据/span>2据/span>-据/span>0.据/span>2据/span> =据/span>1据/span>3.据/span> =据/span>5.据/span>2据/span>+据/span>(据/span>-据/span>1据/span>)据/span>2据/span>+据/span>0.据/span>2据/span> =据/span>2据/span>6.据/span> 。据/span>
然后使用横向产品的幅度定义,我们有据/p>
∥据/span>V.据/span> ×据/span>W.据/span> ∥据/span>=据/span>2据/span>6.据/span> ⋅据/span>1据/span>3.据/span> ⋅据/span>2据/span>2据/span> =据/span>2据/span>2据/span>6.据/span>=据/span>1据/span>3.据/span>。据/span>□据/span>
显示几何解释的等价性和交叉产品的代数解释。据/p>
我们需要表明几何和代数定义给出了相同的幅度和方向。要检查方向,我们将表明两个矢量垂直于据span class="katex"> 一种据/span> 和据span class="katex"> B.据/span> 。这是在几何解释的定义中给出的。对于代数解释,拍摄DOT产品给出:据/p>
(据/span>一种据/span>1据/span>那据/span>一种据/span>2据/span>那据/span>一种据/span>3.据/span>)据/span>⋅据/span>(据/span>一种据/span>2据/span>B.据/span>3.据/span>-据/span>一种据/span>3.据/span>B.据/span>2据/span>那据/span>一种据/span>3.据/span>B.据/span>1据/span>-据/span>一种据/span>1据/span>B.据/span>3.据/span>那据/span>那据/span>一种据/span>1据/span>B.据/span>2据/span>-据/span>一种据/span>2据/span>B.据/span>1据/span>)据/span>(据/span>B.据/span>1据/span>B.据/span>2据/span>那据/span>B.据/span>3.据/span>)据/span>⋅据/span>(据/span>一种据/span>2据/span>B.据/span>3.据/span>-据/span>一种据/span>3.据/span>B.据/span>3.据/span>那据/span>一种据/span>3.据/span>B.据/span>1据/span>-据/span>一种据/span>1据/span>B.据/span>3.据/span>那据/span>那据/span>一种据/span>1据/span>B.据/span>2据/span>-据/span>一种据/span>2据/span>B.据/span>1据/span>)据/span>=据/span>0.据/span>=据/span>0.据/span>。据/span>
此外,自决定簇以来的载体指向相同的方向据/p>
|据/span>|据/span>|据/span>|据/span>|据/span>|据/span>一种据/span>1据/span>B.据/span>1据/span>一种据/span>2据/span>B.据/span>3.据/span>-据/span>一种据/span>3.据/span>B.据/span>3.据/span>一种据/span>2据/span>B.据/span>2据/span>一种据/span>3.据/span>B.据/span>1据/span>-据/span>一种据/span>1据/span>B.据/span>3.据/span>一种据/span>3.据/span>B.据/span>3.据/span>一种据/span>1据/span>B.据/span>2据/span>-据/span>一种据/span>2据/span>B.据/span>1据/span>|据/span>|据/span>|据/span>|据/span>|据/span>|据/span>
沿着第三行扩张是积极的;因此,向量由右手规则确定。据/p>
接下来,通过计算规范的平方来检查矢量的长度是否具有相同的长度。我们有据/p>
∥据/span>一种据/span> ∥据/span>2据/span>∥据/span>B.据/span> ∥据/span>2据/span>罪据/span>2据/span>θ.据/span>=据/span>=据/span>(据/span>一种据/span>1据/span>2据/span>+据/span>一种据/span>2据/span>2据/span>+据/span>一种据/span>3.据/span>2据/span>)据/span>(据/span>B.据/span>1据/span>2据/span>+据/span>B.据/span>2据/span>2据/span>+据/span>B.据/span>3.据/span>2据/span>)据/span>⎝据/span>⎜据/span>⎛据/span>1据/span>-据/span>⎝据/span>⎛据/span>(据/span>一种据/span>1据/span>2据/span>+据/span>一种据/span>2据/span>2据/span>+据/span>一种据/span>3.据/span>2据/span>)据/span>(据/span>B.据/span>1据/span>2据/span>+据/span>B.据/span>2据/span>2据/span>+据/span>B.据/span>3.据/span>2据/span>)据/span> (据/span>一种据/span>1据/span>B.据/span>1据/span>+据/span>一种据/span>2据/span>B.据/span>2据/span>+据/span>一种据/span>3.据/span>B.据/span>3.据/span>)据/span>⎠据/span>⎞据/span>2据/span>⎠据/span>⎟据/span>⎞据/span>一种据/span>1据/span>2据/span>B.据/span>2据/span>2据/span>+据/span>一种据/span>1据/span>2据/span>B.据/span>3.据/span>2据/span>+据/span>一种据/span>2据/span>2据/span>B.据/span>3.据/span>2据/span>+据/span>一种据/span>2据/span>2据/span>B.据/span>1据/span>2据/span>+据/span>一种据/span>3.据/span>2据/span>B.据/span>1据/span>2据/span>+据/span>一种据/span>3.据/span>2据/span>B.据/span>2据/span>2据/span>-据/span>2据/span>一种据/span>1据/span>一种据/span>2据/span>B.据/span>1据/span>B.据/span>2据/span>-据/span>2据/span>一种据/span>2据/span>一种据/span>3.据/span>B.据/span>2据/span>B.据/span>3.据/span>-据/span>2据/span>一种据/span>3.据/span>一种据/span>1据/span>B.据/span>3.据/span>B.据/span>1据/span>。据/span>
对于代数解释,我们有据/p>
=据/span>(据/span>一种据/span>2据/span>B.据/span>3.据/span>-据/span>B.据/span>2据/span>一种据/span>3.据/span>)据/span>2据/span>+据/span>(据/span>-据/span>一种据/span>1据/span>B.据/span>3.据/span>+据/span>B.据/span>1据/span>一种据/span>3.据/span>)据/span>2据/span>+据/span>(据/span>一种据/span>1据/span>B.据/span>2据/span>-据/span>B.据/span>1据/span>一种据/span>2据/span>)据/span>2据/span>(据/span>一种据/span>2据/span>2据/span>B.据/span>3.据/span>2据/span>+据/span>B.据/span>2据/span>2据/span>一种据/span>3.据/span>2据/span>-据/span>2据/span>一种据/span>2据/span>B.据/span>2据/span>一种据/span>3.据/span>B.据/span>3.据/span>)据/span>+据/span>(据/span>一种据/span>1据/span>2据/span>B.据/span>3.据/span>2据/span>+据/span>一种据/span>3.据/span>2据/span>B.据/span>1据/span>2据/span>-据/span>2据/span>一种据/span>3.据/span>一种据/span>1据/span>B.据/span>3.据/span>B.据/span>1据/span>)据/span>+据/span>(据/span>一种据/span>1据/span>2据/span>B.据/span>2据/span>2据/span>+据/span>B.据/span>1据/span>2据/span>一种据/span>2据/span>2据/span>-据/span>2据/span>一种据/span>1据/span>一种据/span>2据/span>B.据/span>1据/span>B.据/span>2据/span>)据/span>。据/span>
通过比较术语,我们看到矢量的长度是相等的。据span class="katex"> □据/span>
我们知道平行四边形的区域是给出的据span class="katex"> (据/span>根据据/span>)据/span>×据/span>(据/span>高度据/span>)据/span>。据/span>在上图中,该区域是大小的产物据span class="katex"> 一种据/span> 和据span class="katex"> B.据/span> :据/span>
区域据/span>=据/span>∥据/span>一种据/span> ∥据/span>∥据/span>∥据/span>∥据/span>B.据/span> ∥据/span>∥据/span>∥据/span>罪据/span>θ.据/span>。据/span>
我们可以看到平行四边形区域等于横向产品的幅度据span class="katex">
(据/span>∥据/span>∥据/span>∥据/span>一种据/span>
×据/span>B.据/span>
∥据/span>∥据/span>∥据/span>=据/span>∥据/span>一种据/span>
∥据/span>∥据/span>∥据/span>∥据/span>B.据/span>
∥据/span>∥据/span>∥据/span>罪据/span>θ.据/span>)据/span>。据/span>也就是说,横向产品的大小等于由两个向量形成的平行四边形区域。据/p>
找到由vectors形成的三角形区域据span class="katex">
一种据/span>=据/span>(据/span>1据/span>那据/span>0.据/span>那据/span>-据/span>1据/span>)据/span>那据span class="katex">
B.据/span>=据/span>(据/span>2据/span>那据/span>1据/span>那据/span>1据/span>)据/span>, 和据span class="katex">
C据/span>=据/span>(据/span>-据/span>1据/span>那据/span>2据/span>那据/span>1据/span>)据/span>。据/span> 我们有据/p>
一种据/span>B.据/span>
一种据/span>C据/span>
=据/span>(据/span>1据/span>那据/span>1据/span>那据/span>2据/span>)据/span>=据/span>(据/span>-据/span>2据/span>那据/span>2据/span>那据/span>2据/span>)据/span>。据/span> 自交叉产品以来据span class="katex">
一种据/span>B.据/span>和据span class="katex">
一种据/span>C据/span>是平行四边形的区域,三角形的面积等于平行四边形区域的一半:据/p>
一种据/span>T.据/span>一种据/span>B.据/span>
×据/span>一种据/span>C据/span>
∥据/span>∥据/span>∥据/span>一种据/span>B.据/span>
×据/span>一种据/span>C据/span>
∥据/span>∥据/span>∥据/span>⇒据/span>一种据/span>T.据/span>=据/span>2据/span>1据/span>∥据/span>∥据/span>∥据/span>一种据/span>B.据/span>
×据/span>一种据/span>C据/span>
∥据/span>∥据/span>∥据/span>=据/span>⎣据/span>⎡据/span>一世据/span>1据/span>-据/span>2据/span>j据/span>1据/span>2据/span>K.据/span>2据/span>2据/span>⎦据/span>⎤据/span>=据/span>(据/span>2据/span>-据/span>4.据/span>)据/span>一世据/span>-据/span>(据/span>2据/span>-据/span>(据/span>-据/span>4.据/span>)据/span>)据/span>j据/span>+据/span>(据/span>2据/span>-据/span>(据/span>-据/span>2据/span>)据/span>)据/span>K.据/span>=据/span>-据/span>2据/span>一世据/span>-据/span>6.据/span>j据/span>+据/span>4.据/span>K.据/span>=据/span>(据/span>-据/span>2据/span>)据/span>2据/span>+据/span>(据/span>-据/span>6.据/span>)据/span>2据/span>+据/span>(据/span>4.据/span>)据/span>2据/span>
=据/span>5.据/span>6.据/span>
=据/span>2据/span>5.据/span>6.据/span>
=据/span>1据/span>4.据/span>
。据/span>□据/span>
表明杂交产品据span class="katex">
一种据/span>
⋅据/span>(据/span>B.据/span>
×据/span>C据/span>
)据/span>=据/span>B.据/span>
⋅据/span>(据/span>C据/span>
×据/span>一种据/span>
)据/span>=据/span>C据/span>
⋅据/span>(据/span>一种据/span>
×据/span>B.据/span>
)据/span>等于平行六面体的体积。据/p>
从横向产品的性质,平行六面体的基部面积是据/p>
一种据/span>根据据/span>=据/span>∥据/span>∥据/span>∥据/span>一种据/span>
×据/span>B.据/span>
∥据/span>∥据/span>∥据/span>。据/span> 让据span class="katex">
θ.据/span>是矢量形成的角度据span class="katex">
C据/span>
用垂直,然后平行的高度是据/p>
高度据/span>=据/span>∥据/span>C据/span>
∥据/span>COS.据/span>θ.据/span>。据/span> 现在,自以来据span class="katex">
一种据/span>
×据/span>B.据/span>
是由矢量形成的平面正常的矢量据span class="katex">
一种据/span>
和据span class="katex">
B.据/span>
那据span class="katex">
θ.据/span>也是vectors之间的角度据span class="katex">
一种据/span>
×据/span>B.据/span>
和据span class="katex">
C据/span>
。所以平行的面积是据/p>
一种据/span>=据/span>一种据/span>根据据/span>×据/span>高度据/span>=据/span>∥据/span>∥据/span>∥据/span>一种据/span>
×据/span>B.据/span>
∥据/span>∥据/span>∥据/span>∥据/span>C据/span>
∥据/span>COS.据/span>θ.据/span>=据/span>(据/span>一种据/span>
×据/span>B.据/span>
)据/span>⋅据/span>C据/span>
。据/span>□据/span>
考虑正常的八边形以原点为中心,如右图所示。八个单位向量从八角形的中心抽取到其每个顶点并在图中标记。对于每对不同的单位向量据span class="katex">
你据/span>
一世据/span>那据/span>你据/span>
j据/span>和据span class="katex">
一世据/span>据据/span>j据/span>,他们的横向产品是计算的。据/p>
所有这些交叉产品的总和是多少?据/p>