10.4 Cross product: a vector orthogonal to two given vectors Cross product of two vectors in space (area of parallelogram) Triple Scalar product (volume of parallelepiped) Torque
The Cross Product: Many applications in physics and engineering involve finding a vector in space that is orthogonal to two given vectors. This vector is called the Cross product. (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.
Vector Products Using Determinants • The cross product can be expressed as • Expanding the determinants gives
Find the cross product for the vectors below. u = <2, 4, 5> and v = <1, -2, -1>
Now that you can do a cross product the next step is to see why this is useful. Let’s look at the 3 vectors from the last problem What is the dot product of ? And ? 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.
Example, You try: 1. Find a unit vector that is orthogonal to both :
Vector Products of Unit Vectors Contrast with scalar products of unit vectors Signs are interchangeable in cross products
If A & B are vectors, their Vector (Cross) Productis defined as: A third vector • The magnitude of vector C is AB sinθwhere θ is the angle between A & B
Vector Product • The magnitude of C, which is AB sinθ is equal to the area of the parallelogram formed by A and B. • The direction of C is perpendicular to the plane formed by A and B • The best way to determine this direction is to use the right-hand rule
Area of a parallelogram = bh, in this diagram, Since 2 vectors in space form a parallelogram h u v
Calculate the area of the triangle where P = (2, 4, -7), Q = (3, 7, 18), and R = (-5, 12, 8)
Now you try! Find the area of the triangle with the given vertices A(1, -4, 3) B(2, 0, 2) C(-2, 2, 0)
Calculate the area of the parallelogram PQRS, where P = (1, 1), Q = (2, 3), R = (5, 4), and S = (4, 2)
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, then find the 4 vectors representing the sides of the Parallelogram, and use the property:
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) z Is the parallelogram a rectangle? y x
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:
Torque • The moment M of a force F about point P M=PQ x F Where magnitude of M measures the tendency of PQ to rotate counterclockwise about axis directed along M.
Pg. 748 Torque problem Vertical force of 50 pounds applied to a 1-ft lever attached to an axle at P. Find the moment of force about P when θ=60.
Homework/Classwork 5-11 odd, 31-37 odd, 41, 42, 45, 46, 47