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Section 9.4: The Cross Product

Section 9.4: The Cross Product. Practice HW from Stewart Textbook (not to hand in) p. 664 # 1, 7-17. Cross Product of Two Vectors. The cross product of two vectors produces a vector (unlike the dot product which produces s scalar) that has important properties. Before

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Section 9.4: The Cross Product

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  1. Section 9.4: The Cross Product Practice HW from Stewart Textbook (not to hand in) p. 664 # 1, 7-17

  2. Cross Product of Two Vectors The cross product of two vectors produces a vector (unlike the dot product which produces s scalar) that has important properties. Before defining the cross product, we first give a method for computing a determinant.

  3. Definition: The determinant of a matrix, denoted by , is defined to be the scalar

  4. Example 1: Compute Solution:

  5. Definition: If a = i + j + k and b = i + j + kbe vectors in 3D space. The cross product is the vector i + j + k

  6. To calculate the cross product more easily without having to remember the formula, we using the following “determinant” form.

  7. We calculate the determinant as follows: (note the alternation in sign)

  8. Example 2: Given the vectors and . Find a.

  9. b.

  10. c.

  11. PropertiesoftheCrossProduct Let a, b, and c be vectors, k be a scalar. • Note! 2. 3. k(kkb) 4. 5.

  12. GeometricPropertiesoftheCrossProduct Let a and b be vectors 1. is orthogonal to both a and b.

  13. Example 3: Given the vectors and , show that the cross product is orthogonal to both a and b.

  14. z Note: , , y x

  15. 2. • if and only a = kb, that is, if the vectors aand b are parallel. 4 gives the area having the vectors aand bas its adjacent sides.

  16. Example 4: Given the points P(0, -2, 0), Q(-1, 3, 4), and R(3,0,6). • Find a vector orthogonal to the plane through these points. b. Find the area of the parallelogram with the vectors and as its adjacent sides. c. Find area of the triangle PQR. Solution: (In the typewritten notes)

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