blocks|key|414410|text|实现1返回将由输入向量的规则3D叉积产生的向量的大小，隐含地将其Z值取为0(即，将2D空间视为3D空间中的平面)。3D叉积将垂直于该平面，因此具有0+X&Y分量(因此返回的标量是3D叉积向量的Z值)。|type|unstyled|depth|inlineStyleRanges|entityRanges|data|414411|请注意，由3D叉积产生的向量的大小也等于两个向量之间的平行四边形的面积，这给了实现1另一个目的。此外，此区域是有符号的，可用于确定从V1旋转到V2是逆时针还是逆时针移动。还应注意，实现1是从这两个向量构建的2x2矩阵的行列式。|414412|实现2返回与输入矢量垂直的矢量，该矢量仍然在相同的2D平面中。不是经典意义上的叉积，而是一致的“给我一个垂直向量”。|414413|请注意，在叉积操作下，3D欧几里得空间是封闭的--即，两个3D向量的叉积返回另一个3D向量。上述两种2D实现在某种程度上都与之不一致。|414414|希望这能帮到你。|414415|entityMap^0|0|0|0|0|0^^$0|@$1|2|3|4|5|6|7|L|8|@]|9|@]|A|$]]|$1|B|3|C|5|6|7|M|8|@]|9|@]|A|$]]|$1|D|3|E|5|6|7|N|8|@]|9|@]|A|$]]|$1|F|3|G|5|6|7|O|8|@]|9|@]|A|$]]|$1|H|3|I|5|6|7|P|8|@]|9|@]|A|$]]|$1|J|3|-4|5|6|7|Q|8|@]|9|@]|A|$]]]|K|$]]

Implementation 1 returns the magnitude of the vector that would result from a regular 3D cross product of the input vectors, taking their Z values implicitly as 0 (i.e. treating the 2D space as a plane in the 3D space). The 3D cross product will be perpendicular to that plane, and thus have 0 X &amp; Y components (thus the scalar returned is the Z value of the 3D cross product vector).

Note that the magnitude of the vector resulting from 3D cross product is also equal to the area of the parallelogram between the two vectors, which gives Implementation 1 another purpose. In addition, this area is signed and can be used to determine whether rotating from V1 to V2 moves in an counter clockwise or clockwise direction. It should also be noted that implementation 1 is the determinant of the 2x2 matrix built from these two vectors.

Implementation 2 returns a vector perpendicular to the input vector still in the same 2D plane. Not a cross product in the classical sense but consistent in the "give me a perpendicular vector" sense.

Note that 3D euclidean space is closed under the cross product operation--that is, a cross product of two 3D vectors returns another 3D vector. Both of the above 2D implementations are inconsistent with that in one way or another.

Hope this helps...

blocks|key|414673|text|简而言之：：，这是一个数学技巧的简写符号。|type|unstyled|depth|inlineStyleRanges|offset|length|style|BOLD|entityRanges|data|414674|长篇解释：|414675|你不能对二维空间中的向量做叉积。该操作没有在那里定义。|414676|然而，通常有趣的是评估两个向量的叉积，假设通过将它们的z坐标设置为零来将2D向量扩展到3D向量。这与在xy平面上使用3D矢量相同。|414677|如果您以这种方式扩展向量并计算这种扩展向量对的叉积，您将注意到只有z分量具有有意义的值:x和y始终为零。|414678|这就是为什么结果的z分量通常简单地作为标量返回的原因。例如，这个标量可以用来寻找2D空间中三个点的缠绕。|414679|从纯数学的角度来看，2D空间中的叉积是不存在的，标量版本是hack，返回2D向量的2D叉积根本没有意义。|414680|entityMap^0|5|1|0|0|5|0|0|0|0|0|0^^$0|@$1|2|3|4|5|6|7|T|8|@$9|U|A|V|B|C]]|D|@]|E|$]]|$1|F|3|G|5|6|7|W|8|@$9|X|A|Y|B|C]]|D|@]|E|$]]|$1|H|3|I|5|6|7|Z|8|@]|D|@]|E|$]]|$1|J|3|K|5|6|7|10|8|@]|D|@]|E|$]]|$1|L|3|M|5|6|7|11|8|@]|D|@]|E|$]]|$1|N|3|O|5|6|7|12|8|@]|D|@]|E|$]]|$1|P|3|Q|5|6|7|13|8|@]|D|@]|E|$]]|$1|R|3|-4|5|6|7|14|8|@]|D|@]|E|$]]]|S|$]]

In short: It's a shorthand notation for a mathematical hack.

Long explanation:

You can't do a cross product with vectors in 2D space. The operation is not defined there.

However, often it is interesting to evaluate the cross product of two vectors assuming that the 2D vectors are extended to 3D by setting their z-coordinate to zero. This is the same as working with 3D vectors on the xy-plane. 

If you extend the vectors that way and calculate the cross product of such an extended vector pair you'll notice that only the z-component has a meaningful value: x and y will always be zero.

That's the reason why the z-component of the result is often simply returned as a scalar. This scalar can for example be used to find the winding of three points in 2D space.

From a pure mathematical point of view the cross product in 2D space does not exist, the scalar version is the hack and a 2D cross product that returns a 2D vector makes no sense at all.

blocks|key|414702|text|叉积的另一个有用的属性是它的大小与两个向量之间的角度的正弦有关：|type|unstyled|depth|inlineStyleRanges|entityRanges|data|414703|414704|+%7Ca+x+b%7C=+%7Ca%7C。%7Cb%7C。正弦(θ)|blockquote|414705|414706|414707|或|414708|414709|正弦(θ)=%7Ca+x+b%7C/+(%7Ca%7C+.%7Cb%7C)|414710|414711|414712|因此，在上面的实现1中，如果预先知道a和b是单位向量，那么该函数的结果就是那个sine()值。|offset|length|style|CODE|414713|entityMap^0|0|0|0|0|0|0|0|0|0|0|I|1|K|1|0^^$0|@$1|2|3|4|5|6|7|W|8|@]|9|@]|A|$]]|$1|B|3|-4|5|6|7|X|8|@]|9|@]|A|$]]|$1|C|3|D|5|E|7|Y|8|@]|9|@]|A|$]]|$1|F|3|-4|5|6|7|Z|8|@]|9|@]|A|$]]|$1|G|3|-4|5|6|7|10|8|@]|9|@]|A|$]]|$1|H|3|I|5|6|7|11|8|@]|9|@]|A|$]]|$1|J|3|-4|5|6|7|12|8|@]|9|@]|A|$]]|$1|K|3|L|5|E|7|13|8|@]|9|@]|A|$]]|$1|M|3|-4|5|6|7|14|8|@]|9|@]|A|$]]|$1|N|3|-4|5|6|7|15|8|@]|9|@]|A|$]]|$1|O|3|P|5|6|7|16|8|@$Q|17|R|18|S|T]|$Q|19|R|1A|S|T]]|9|@]|A|$]]|$1|U|3|-4|5|6|7|1B|8|@]|9|@]|A|$]]]|V|$]]

Another useful property of the cross product is that its magnitude is related to the sine of the angle between the two vectors:

<blockquote>
 | a x b | = |a| . |b| . sine(theta)
</blockquote>

or

<blockquote>
 sine(theta) = | a x b | / (|a| . |b|)
</blockquote>

So, in implementation 1 above, if <code>a</code> and <code>b</code> are known in advance to be unit vectors then the result of that function is exactly that sine() value.

blocks|key|3469119|text|我在我的计算中使用二维叉积来找到一个物体的新的正确的旋转，这个物体是由一个力矢量作用的，相对于它的质心的任意点。(标量Z一。)|type|unstyled|depth|inlineStyleRanges|entityRanges|data|3469120|entityMap^0|0^^$0|@$1|2|3|4|5|6|7|D|8|@]|9|@]|A|$]]|$1|B|3|-4|5|6|7|E|8|@]|9|@]|A|$]]]|C|$]]

I'm using 2d cross product in my calculation to find the new correct rotation for an object that is being acted on by a force vector at an arbitrary point relative to its center of mass. (The scalar Z one.)

blocks|key|414736|text|实现1是两个向量的perp点积。据我所知，2D图形最好的参考资料是优秀的Graphics+Gems系列。如果你正在做scratch+2D的工作，拥有这些书是非常重要的()。第四卷有一篇名为“Pleasures+of+Perp+Dot+Products”的文章，详细介绍了它的许多用途。|type|unstyled|depth|inlineStyleRanges|offset|length|style|BOLD|entityRanges|data|414737|perp点积的一个主要用途是获得两个向量之间的角度的缩放sin，就像点积返回角度的缩放cos一样。当然，您可以一起使用点积和perp点积来确定两个向量之间的角度。|CODE|414738|Here是关于它的一篇文章，here是Wolfram+Math+World的文章。|414739|entityMap|0|LINK|mutability|MUTABLE|url|http://www.graphicsgems.org/|1|http://johnblackburne.blogspot.co.il/2012/02/perp-dot-product.html|2|http://mathworld.wolfram.com/PerpDotProduct.html^0|9|6|10|D|0|0|0|6|Y|2|1N|2|1Q|6|S|3|17|3|0|0|4|1|E|4|2|0^^$0|@$1|2|3|4|5|6|7|W|8|@$9|X|A|Y|B|C]]|D|@$9|Z|A|10|1|11]]|E|$]]|$1|F|3|G|5|6|7|12|8|@$9|13|A|14|B|C]|$9|15|A|16|B|C]|$9|17|A|18|B|C]|$9|19|A|1A|B|C]|$9|1B|A|1C|B|H]|$9|1D|A|1E|B|H]]|D|@]|E|$]]|$1|I|3|J|5|6|7|1F|8|@]|D|@$9|1G|A|1H|1|1I]|$9|1J|A|1K|1|1L]]|E|$]]|$1|K|3|-4|5|6|7|1M|8|@]|D|@]|E|$]]]|L|$M|$5|N|O|P|E|$Q|R]]|S|$5|N|O|P|E|$Q|T]]|U|$5|N|O|P|E|$Q|V]]]]

Implementation 1 is the perp dot product of the two vectors. The best reference I know of for 2D graphics is the excellent <a href="http://www.graphicsgems.org" rel="noreferrer">Graphics Gems</a> series. If you're doing scratch 2D work, it's really important to have these books. Volume IV has an article called "The Pleasures of Perp Dot Products" that goes over a lot of uses for it.

One major use of perp dot product is to get the scaled <code>sin</code> of the angle between the two vectors, just like the dot product returns the scaled <code>cos</code> of the angle. Of course you can use dot product and perp dot product together to determine the angle between two vectors.

<a href="http://johnblackburne.blogspot.co.il/2012/02/perp-dot-product.html" rel="noreferrer">Here</a> is a post on it and <a href="http://mathworld.wolfram.com/PerpDotProduct.html" rel="noreferrer">here</a> is the Wolfram Math World article.

blocks|key|3255967|text|一个有用的2D向量运算是返回标量的叉积。我使用它来查看多边形中的两个连续边是向左还是向右弯曲。|type|unstyled|depth|inlineStyleRanges|entityRanges|data|3255968|从Chipmunk2D源文件中：|offset|length|3255969|///+2D+vector+cross+product+analog.
///+The+cross+product+of+2D+vectors+results+in+a+3D+vector+with+only+a+z+component.
///+This+function+returns+the+magnitude+of+the+z+value.
static+inline+cpFloat+cpvcross(const+cpVect+v1,+const+cpVect+v2)
{
++++++++return+v1.x*v2.y+-+v1.y*v2.x;
}|code-block|syntax|javascript|3255970|entityMap|0|LINK|mutability|MUTABLE|url|https://chipmunk-physics.net/^0|0|1|A|0|0|0^^$0|@$1|2|3|4|5|6|7|S|8|@]|9|@]|A|$]]|$1|B|3|C|5|6|7|T|8|@]|9|@$D|U|E|V|1|W]]|A|$]]|$1|F|3|G|5|H|7|X|8|@]|9|@]|A|$I|J]]|$1|K|3|-4|5|6|7|Y|8|@]|9|@]|A|$]]]|L|$M|$5|N|O|P|A|$Q|R]]]]

A useful 2D vector operation is a cross product that returns a scalar. I use it to see if two successive edges in a polygon bend left or right.

From the <a href="https://chipmunk-physics.net/" rel="nofollow noreferrer">Chipmunk2D</a> source:

<pre><code>/// 2D vector cross product analog.
/// The cross product of 2D vectors results in a 3D vector with only a z component.
/// This function returns the magnitude of the z value.
static inline cpFloat cpvcross(const cpVect v1, const cpVect v2)
{
 return v1.x*v2.y - v1.y*v2.x;
}
</code></pre>

From wikipedia: 

<blockquote>
 the cross product is a binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which is perpendicular to the plane containing the two input vectors.
</blockquote>

Given that the definition is only defined in three (<a href="https://en.wikipedia.org/wiki/Seven-dimensional_cross_product#Consequences_of_the_defining_properties" rel="noreferrer">or seven, one and zero</a>) dimensions, how does one calculate the cross product of two 2d vectors?

I have seen two implementations. One returns a new vector (but only accepts a single vector), the other returns a scalar (but is a calculation between two vectors).

Implementation 1 (returns a scalar):

<pre><code>float CrossProduct(const Vector2D &amp; v1, const Vector2D &amp; v2) const
{
 return (v1.X*v2.Y) - (v1.Y*v2.X);
}
</code></pre>

Implementation 2 (returns a vector):

<pre><code>Vector2D CrossProduct(const Vector2D &amp; v) const
{
 return Vector2D(v.Y, -v.X);
}
</code></pre>

Why the varying implementations? What would I use the scalar implementation for? What would I use the vector implementation for?

The reason I ask is because I'm writing a Vector2D class myself and don't know which method to use.

Calculating a 2D Vector's Cross Product

翻译质量差，导致语言生硬或混乱。

没有提供实际的解决方法或示例。

解答不清晰，无法理解或解决问题。

页面排版不美观，阅读体验差。

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来自维基百科：叉积是在three-dimensional 空间中对两个向量进行的二元运算，得到另一个垂直于包含两个输入向量的平面的向量。假设定义只在三维()维度上定义，如何计算两个2d向量的叉积？我已经看到了两个实现。一个返回新的向量(但只接受单个向量)，另一个返回标量(但是两个向量之间的计算)。实现1(返回标量)：f...

问计算二维矢量的叉积
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