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MAT 1236 Calculus III. Section 12.4 The Cross Product. http://myhome.spu.edu/lauw. HW…. WebAssign 12.4 Read 12.5 ( Seriously! ): The first not too easy section in Calculus. Preview. Define a new operation on vectors: The Cross Product

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Mat 1236 calculus iii

MAT 1236Calculus III

Section 12.4

The Cross Product

http://myhome.spu.edu/lauw


Mat 1236 calculus iii
HW…

  • WebAssign 12.4

  • Read 12.5 (Seriously!): The first not too easy section in Calculus


Preview
Preview

  • Define a new operation on vectors: The Cross Product

  • Unlike the dot product, the cross product of two vectors is a vector.

  • Properties of the cross product.



We are interested in
We are Interested in …

  • Given 2 vectors, they “span” a plane

  • Find a vector perpendicular to this plane


The cross product
The Cross Product

If and , the cross product of a and b is the vector


The cross product1
The Cross Product

The formula is traditionally memorized by using (formal) determinant expansions







The cross product2
The Cross Product

The formula is traditionally memorized by using (formal) determinant expansions



Expectations
Expectations

  • You are expected to use the above standard procedure to find the cross product.

  • You are expected to show all the steps. Keep in mind, good practices are key to minimize the chance of making mistakes.




Property b1
Property B

In addition, the cross product obeys the Right Hand Rule.







In particular1
In Particular

is in the same direction of k and


Property d
Property D

Two nonzero vectors and are parallel if and only if


Property d why
Property D (Why?)

Two nonzero vectors and are parallel if and only if


Property e
Property E

The length of the cross product axb is equal to the area of the parallelogram

determined by a and b.


Example 2
Example 2

Find a vector perpendicular to the plane that passes through the points

P(6,0,0) , Q(1,1,1), R(0,0,2)


Example 3
Example 3

Find the area of the triangle with vertices

P(6,0,0) , Q(1,1,1), R(0,0,2)


Other properties
Other Properties

Right Hand Rule

Default

Reference only