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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 1236Calculus 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 • Unlike the dot product, the cross product of two vectors is a vector. • Properties of the cross product.
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
The Cross Product The formula is traditionally memorized by using (formal) determinant expansions
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 B In addition, the cross product obeys the Right Hand Rule.
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 Right Hand Rule Default Reference only