Calculating a 2D Vector s Cross Product

Question

From wikipedia       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    Given that the definition is only defined in three  or seven  one and zero  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    float CrossProduct const Vector2D  amp  v1  const Vector2D  amp  v2  const       return  v1 X v2 Y  -  v1 Y v2 X       Implementation 2  returns a vector    Vector2D CrossProduct const Vector2D  amp  v  const       return Vector2D v Y  -v X       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

User · Answer

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

User · Answer

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

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Another useful property of the cross product is that its magnitude is related to the sine of the angle between the two vectors        a x b      a     b    sine theta    or     sine theta      a x b       a     b     So  in implementation 1 above  if a and b are known in advance to be unit vectors then the result of that function is exactly that sine   value

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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

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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 Chipmunk2D source       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

User · Answer

Another useful property of the cross product is that its magnitude is related to the sine of the angle between the two vectors        a x b      a     b    sine theta    or     sine theta      a x b       a     b     So  in implementation 1 above  if a and b are known in advance to be unit vectors then the result of that function is exactly that sine   value

User · Answer

Another useful property of the cross product is that its magnitude is related to the sine of the angle between the two vectors        a x b      a     b    sine theta    or     sine theta      a x b       a     b     So  in implementation 1 above  if a and b are known in advance to be unit vectors then the result of that function is exactly that sine   value

User · Answer

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

User · Answer

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 Chipmunk2D source       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

User · Answer

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

User · Answer

Another useful property of the cross product is that its magnitude is related to the sine of the angle between the two vectors        a x b      a     b    sine theta    or     sine theta      a x b       a     b     So  in implementation 1 above  if a and b are known in advance to be unit vectors then the result of that function is exactly that sine   value

User · Answer

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

User · Answer

Implementation 1 is the perp dot product of the two vectors   The best reference I know of for 2D graphics is the excellent Graphics Gems 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 sin of the angle between the two vectors  just like the dot product returns the scaled cos of the angle   Of course you can use dot product and perp dot product together to determine the angle between two vectors   Here is a post on it and here is the Wolfram Math World article

User · Answer

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

User · Answer

Implementation 1 is the perp dot product of the two vectors   The best reference I know of for 2D graphics is the excellent Graphics Gems 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 sin of the angle between the two vectors  just like the dot product returns the scaled cos of the angle   Of course you can use dot product and perp dot product together to determine the angle between two vectors   Here is a post on it and here is the Wolfram Math World article

User · Answer

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

User · Answer

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

User · Answer

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

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