MAT 1236 Calculus III Section 12.4 The Cross Product

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Presentation transcript:

MAT 1236 Calculus III Section 12.4 The Cross Product

HW… WebAssign 12.4 (18 problems, 98 min.) Read 12.5 (Seriously!): The first not-too- easy section in Calculus

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.

Classwork of the last section... We did not have time to work on the last classwork....

The Right Hand Rule FBI

We are Interested in … Given 2 vectors, they “span” a plane Find a vector perpendicular to this plane

The Cross Product If and, the cross product of a and b is the vector

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

2x2 Determinant Expansions

3x3 Determinant Expansions

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

Example 1

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 A

Property B

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

Property B (Why?)

Example 1 (Verify Property B)

Property C

Property C (Why?)

In Particular

is in the same direction of k and

Property D Two nonzero vectors and are parallel if and only if

Property D (Why?) Two nonzero vectors and are parallel if and only if

Property E The length of the cross product axb is equal to the area of the parallelogram determined by a and b.

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 Find the area of the triangle with vertices P(6,0,0), Q(1,1,1), R(0,0,2)

Other Properties Reference only Default Right Hand Rule