10.4 Cross product: a vector orthogonal to two given vectors

1. 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

3. Cofactor Expansion

4. 2nd method (not in book)

5. 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.

6. Vector Products Using Determinants • The cross product can be expressed as • Expanding the determinants gives

7. Find the cross product for the vectors below. u = <2, 4, 5> and v = <1, -2, -1>

8. 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.

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

10. Vector Products of Unit Vectors Contrast with scalar products of unit vectors Signs are interchangeable in cross products

13. 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

14. 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

15. Area of a parallelogram = bh, in this diagram, Since 2 vectors in space form a parallelogram h u v

16. Calculate the area of the triangle where P = (2, 4, -7), Q = (3, 7, 18), and R = (-5, 12, 8)

17. 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)

18. Calculate the area of the parallelogram PQRS, where P = (1, 1), Q = (2, 3), R = (5, 4), and S = (4, 2)

19. 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:

20. 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

21. 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

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

23. 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.

24. 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.

25. Homework/Classwork 5-11 odd, 31-37 odd, 41, 42, 45, 46, 47

26. TI 89 Graphing Calculator