text: CROSS[1]
  
  # CROSS[1] - THE CROSS PRODUCT We met this before when we...
  
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  THE CROSS PRODUCT We met this before when we studied torques! The geometrical definition of a cross product is AB x CB C s i n   where θ is the angle FROM as shown in the drawing: BtoC OK, so far, so good. This gives us the magnitude, but not the direction. For this, we need to use our “circle”. B C θ ˆ x ˆ y ˆ z
  
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  Let’s do a simple problem:  ˆˆ A3 x2 z B 4 y2 z What is A B? A B 3x 2z 4y 2z In elementary algebra, we learned the F.O.I.L. rule for multiplying two binomials together: F.O.I.L stands for First, Outside, Inside, Last. The           cross product becomes: AB 3 4xy 3 2xz 2 4zy 2 2zz AB 1 2xy 6xz 8zy 4zz To do the cross products of the unit vectors in the square brackets, we look to the circle! Put yo              
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explanation: The document is an educational text that introduces and explains the concept of the cross product in mathematics, specifically in the context of vector algebra. It provides a geometrical definition of the cross product and explains how to determine both the magnitude and direction of the cross product. The text also includes a simple problem that demonstrates how to calculate the cross product of two vectors using the F.O.I.L. rule and the unit circle for direction.
  
  The document contains a question and an answer to that question.
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: CROSS[1]
  
  # CROSS[1] - THE CROSS PRODUCT We met this before when we...
  
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  THE CROSS PRODUCT We met this before when we studied torques! The geometrical definition of a cross product is AB x CB C s i n   where θ is the angle FROM as shown in the drawing: BtoC OK, so far, so good. This gives us the magnitude, but not the direction. For this, we need to use our “circle”. B C θ ˆ x ˆ y ˆ z
  
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  View Full Document
  Let’s do a simple problem:  ˆˆ A3 x2 z B 4 y2 z What is A B? A B 3x 2z 4y 2z In elementary algebra, we learned the F.O.I.L. rule for multiplying two binomials together: F.O.I.L stands for First, Outside, Inside, Last. The           cross product becomes: AB 3 4xy 3 2xz 2 4zy 2 2zz AB 1 2xy 6xz 8zy 4zz To do the cross products of the unit vectors in the square brackets, we look to the circle! Put yo              
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distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is an educational text that introduces and explains the concept of the cross product in mathematics, specifically in the context of vector algebra. It provides a geometrical definition of the cross product and explains how to determine both the magnitude and direction of the cross product. The text also includes a simple problem that demonstrates how to calculate the cross product of two vectors using the F.O.I.L. rule and the unit circle for direction.
  
  The document contains a question and an answer to that question.
  
  Label: question-and-answer