0.4 Cross Product Study Book p 18 , Stewart 9.4

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0.4 Cross Product Study Book p 18 ,
Stewart 9.4 Appendix A, Grossman 3.4
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• Dot product yields a scalar
• that tells us about the angle between 2 vectors.
• Cross product gives a vector
• that is perpendicular to the two vectors crossed
  
• ie normal or orthogonal to those crossed.

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• Definition of Cross Product
• (also called vector product)
• (p 183 of Grossman, App A)
• u x v ( (b1c2 - c1b2) , - (a1c2 - c1 a2)
  , (a1 b2 - b1 a2) )
• (If you have studied determinants, use that
  notation.)
• If not, cross product is easy to find if you
  write the 2nd vector below the 1st
• a1 b1 c1
• a2 b2 c2
• Block out 1 column at a time use a cross to
  multiply subtract. But note the change of sign
  for the middle term!
• Th 2 of 3.4 in Grossman notes important
  properties .

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• Th 2 3 yield its length and direction
• Th 2 vi proves
• cross product gives a vector perpendicular
• to the two vectors crossed.
• Th 3 gives its length u x v u v
  sin t
• Note how similar this is to dot
  product.
• It can be shown that its direction is that of the
  thumb if we use the right-hand rule.

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• Another application area volume
• The area of the parallelogram with adjacent sides
  formed by vectors u v is
• area u v sin t u x
  v
• The area of a triangle is half of that.
• The volume of the parallelepiped defined by the
  edges u, v w is vol w . ( u x v
  )
• It is the absolute value of the scalar triple
  product taken in any order
• v . ( w x u ) or u . (v x w) )

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Homework Read Study Book 0.4
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• Grossman Ch 3.4
• Master Q 1-30, 37, 38.
• Prove Th 2 ii vi, by writing out components.
• Write solutions to Q 5, 21, 23, 27, 37.
• Stewart, Ch 9.4 Master 1, 3, 7-18.
• Browse Vectors in H-Hallett et al if you have it
• Ch 12.1 - 12.4 of 2nd Ed, or
• Ch 13.1 - 13.4 of 3rd Ed.

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Objectives

• Use cross product to find a vector perpendicular
  to those crossed.
• Know the properties listed in Grossman, p 185.
• Be aware of similarities with dot product,
• differences.
• Use cross product to find areas of parallelograms
  triangles volumes of parallelepipeds.