The Cross Product of Two Vectors In Space Section

Precalculus Extra Credit Proj. Due 4/30

HWQ 4/17/2013 Find a vector in the same direction as v that has a magnitude of 2.

Just as 2 points in space determine a vector, 2 vectors in space determine a plane. Many applications in physics and engineering involve finding a vector in space that is orthogonal to two given vectors. In this lesson you will learn: How to find the cross product of 2 vectors in space. Applications of the cross product. How to find the triple scalar product. Applications of the triple scalar product.

The Cross Product: A vector in space that is orthogonal to two given vectors. (Note: that while the dot product was a scalar, the cross product is a vector.) The cross product of u and v is the vector u x v. The cross product of two vectors, unlike the dot product, represents a vector. A convenient way to find u x v is to use a determinant involving vector u and vector v. The cross product is found by taking this determinant.

Find the cross product for the vectors below, then find the magnitude of the cross product. u x v and v x u have equal lengths in opposite directions. Both u x v and v x u are perpendicular to the plane determined by u and v.

Let’s look at the 3 vectors from the last problem What is the dot product of And ? If you answered 0 in both cases, you would be correct. Recall that whenever two non-zero vectors are perpendicular, their dot product is 0. Thus the cross product creates a vector perpendicular to the vectors u and v. 0 0 ?

Algebraic Properties of the Cross Product:

Example, You try: 1. Find a unit vector that is orthogonal to both :

u v Area of a parallelogram = bh, in this diagram, area= h Since 2 vectors in space form a parallelogram A geometric property of the cross product is:

Geometric Properties of the Cross Product :

Geometric application example. You try: Find the area of the triangle with the given vertices. The area A of the triangle having u and v as adjacent sides is given by: To begin, create 2 vectors representing adjacent sides of the triangle. Make sure they have the same initial point. A(1,-4,3) B(2,0,2) C(-2,2,0)

Geometric application example: Show that the quadrilateral with vertices at the following points is a parallelogram. Find the area of the parallelogram. Is the parallelogram a rectangle? A(5,2,0) B(2,6,1) C(2,4,7) D(5,0,6) To begin, plot the vertices below, then find the 4 vectors representing the sides of the Parallelogram, and use the property:

The angle between two vectors in space can be found using the following geometric property of the cross product: The angle between 2 vectors in space:

Show that the quadrilateral with vertices at the following points is a parallelogram. Find the area of the parallelogram. Is the parallelogram a rectangle? A(5,2,0) B(2,6,1) C(2,4,7) D(5,0,6) x y z Is the parallelogram a rectangle?

Triple Scalar Product: For the vectors u, v, and w in space, the dot product of u and is called the triple scalar product of u, v, and w. A Geometric property of the triple scalar product: The volume V of a parallelepiped with vectors u, v, and w as adjacent edges is given by: A parallelepiped is a figure created when a parallelogram has depth

Example. You Try: 1. Find the volume of a parallelepiped having adjacent edges:

Homework: Day 1: Pg odd

Homework: Day 2: Pg even

A note about the differing angles between 2 vectors depending on which formula is used: In the quadrilateral example, when the sine formula is used to find the angle between the vectors,, yet when the cosine formula is used, the The angle found with the cosine formula is 180-the other angle. Using a positive value for the dot product in the cosine formula yields the same angle. So they either find the same angle, or the angle of one vector with the opposite of the other vector.

The dot product is the scalar product. Scalar product means the product value is the scalar quantity which is obtained by the cosine angle between two vectors and the cross product means the vector product which implies that the product value is vector quantity. The vector quantity is obtained by the sine angle between the two vectors Cos theta vs. sine theta

#46) Angle = 60 degrees Vector in xz plane z-axis vector Use angle between 2 vectors: Solve it for x: x is the larger of the values Vector=