The Cross Product

1
The Cross Product

• Third Type of Multiplying Vectors

2
Cross Products
3
Determinants

• It is much easier to do this using determinants
  because we do not have to memorize a formula.
• Determinants were used last year when doing
  matrices
• Remember that you multiply each number across and
  subtract their products

4
Finding Cross Products Using Equation
5
Evaluating a Determinant
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Evaluating Determinants
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Using Determinants to Find Cross Products

• This concept can help us find cross products.
• Ignore the numbers included in the column under
  the vector that will be inserted when setting up
  the determinant.

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Using Determinants to Find Cross Products

• Find v x w given
• v i j
• w 2i j k

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Using Determinants to Find Cross Products
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Using Determinants to Find Cross Products

• If v 2i 3j 5k and w i 2j 3k,
• find
• (a) v x w
• (b) w x v
• (c) v x v

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Using Determinants to Find Cross Products
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Using Determinants to Find Cross Products
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Using Determinants to Find Cross Products
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Algebraic Properties of the Cross Product

• If u, v, and w are vectors in space and if a is a
  scalar, then
• u x u 0
• u x v -(v x u)
• a(u x v) (au) x v u x (av)
• u x (v w) (u x v) (u x w)

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Examples

• Given u 2i 3j k v -3i 3j 2k
• w i j 3k
• Find
• (a) (3u) x v
• (b) v . (u x w)

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Examples
17
Examples
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Geometric Properties of the Cross Product

• Let u and v be vectors in space
• u x v is orthogonal to both u and v.
• u x v u v sin q, where q is the
  angle between u and v.
• u x v is the area of the parallelogram having
  u ? 0 and v ? 0 as adjacent sides

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Geometric Properties of the Cross Product

• u x v 0 if and only if u and v are parallel.

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Finding a Vector Orthogonal to Two Given Vectors

• Find a vector that is orthogonal to
• u 2i 3j k and v i j 3k
• According to the preceding slide, u x v is
  orthogonal to both u and v. So to find the vector
  just do u x v

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Finding a Vector Orthogonal to Two Given Vectors
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Finding a Vector Orthogonal to Two Given Vectors

• To check to see if the answer is correct, do a
  dot product with one of the given vectors.
  Remember, if the dot product 0 the vectors are
  orthogonal

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Finding a Vector Orthogonal to Two Given Vectors
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Finding the Area of a Parallelogram

• Find the area of the parallelogram whose vertices
  are P1 (0, 0, 0),
• P2 (3,-2, 1), P3 (-1, 3, -1) and
• P4 (2, 1, 0)
• Two adjacent sides of this parallelogram are u
  P1P2 and v P1P3.

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Finding the Area of the Parallelogram
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Your Turn

• Try to do page 653 problems 1 47 odd.