text: # Why is cross product in 3D?
  
  ## Why is cross product in 3D?
  
  Cross product vs. The dot product works in any number of dimensions, but the cross product only works in 3D. The dot product measures how much two vectors point in the same direction, but the cross product measures how much two vectors point in different directions.
  
  What is the cross product of 3D vectors?
  
  The cross product of two 3D vectors is another vector in the same 3D vector space. Since the result is a vector, we must specify both the length and the direction of the resulting vector: length(a Γ b) = |a Γ b| = |a| |b| sinΞ
  
  How do you do cross product in two dimensions?
  
  The cross product of two collinear vectors is zero, and so β π΄ Γ β π΄ = 0 . The area of the parallelogram spanned by β π΄ and β π΅ is given by β β β π΄ Γ β π΅ β β . It follows that the area of the triangle with β π΄ and β π΅ defining two of its sides is given by 1 2 β β β π΄ Γ β π΅ β β .
  
  ### What does the 2d cross product represent?
  
  I will also note that the “2D cross product” is also commonly referred to as the “perpendicular dot product” or “perp dot product”: the dot product of the CCW perpendicular of A with the (original) B. By “CCW perpendicular”, I mean the vector 90 degrees counterclockwise; the CCW perpendicular of (x, y) is (-y, x).
  
  Is cross product only in r3?
  
  A 2-fold cross vector exists in dimension 3 and 7. Therefore, the “bilinear” cross product can only exists with two factors in 3D and 7D. The 3D cross product is well known, the 7D cross product can be found (both in coordinate and free coordinate versions) in wikipedia.
  
  Can cross product be used in 2d?
  
  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.
  
  ## What is the cross product of two dimensional vectors?
  
  The cross product a Γ b is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.
  
  Can cross product have two dimensional vectors?
  
  What is the cross product equal to?
  
  The cross product of two vectors, say A Γ B, is equal to another vector at right angles to both, and it happens in the three-dimensions.
explanation: The document contains multiple questions and answers related to the cross product in different dimensions. It discusses the properties and applications of the cross product in 3D and 2D, and explains why the cross product is only defined in 3D and 7D.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Why is cross product in 3D?
  
  ## Why is cross product in 3D?
  
  Cross product vs. The dot product works in any number of dimensions, but the cross product only works in 3D. The dot product measures how much two vectors point in the same direction, but the cross product measures how much two vectors point in different directions.
  
  What is the cross product of 3D vectors?
  
  The cross product of two 3D vectors is another vector in the same 3D vector space. Since the result is a vector, we must specify both the length and the direction of the resulting vector: length(a Γ b) = |a Γ b| = |a| |b| sinΞ
  
  How do you do cross product in two dimensions?
  
  The cross product of two collinear vectors is zero, and so β π΄ Γ β π΄ = 0 . The area of the parallelogram spanned by β π΄ and β π΅ is given by β β β π΄ Γ β π΅ β β . It follows that the area of the triangle with β π΄ and β π΅ defining two of its sides is given by 1 2 β β β π΄ Γ β π΅ β β .
  
  ### What does the 2d cross product represent?
  
  I will also note that the “2D cross product” is also commonly referred to as the “perpendicular dot product” or “perp dot product”: the dot product of the CCW perpendicular of A with the (original) B. By “CCW perpendicular”, I mean the vector 90 degrees counterclockwise; the CCW perpendicular of (x, y) is (-y, x).
  
  Is cross product only in r3?
  
  A 2-fold cross vector exists in dimension 3 and 7. Therefore, the “bilinear” cross product can only exists with two factors in 3D and 7D. The 3D cross product is well known, the 7D cross product can be found (both in coordinate and free coordinate versions) in wikipedia.
  
  Can cross product be used in 2d?
  
  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.
  
  ## What is the cross product of two dimensional vectors?
  
  The cross product a Γ b is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.
  
  Can cross product have two dimensional vectors?
  
  What is the cross product equal to?
  
  The cross product of two vectors, say A Γ B, is equal to another vector at right angles to both, and it happens in the three-dimensions.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains multiple questions and answers related to the cross product in different dimensions. It discusses the properties and applications of the cross product in 3D and 2D, and explains why the cross product is only defined in 3D and 7D.
  
  Label: multiple-question-and-answer