Cross Product

1. Math 200 Week 2 - Monday Cross Product

2. Math 200 How is the cross product defined for vectors? How does it interact with other operations on vectors? What uses are there for the cross product? Main Questions for Today

3. We define the cross product of two vectors in the following way… v x w is a vector orthogonal to both v and w consistent with the right-hand rule ||v x w|| is the area of the parallelogram with adjacent sides v and w Math 200 Definition

4. Math 200 True, but before doing that we need another mathematical tool called the determinant, which comes to use from linear algebra (the study of matrices). A matrix is just an array of numbers. • They can be rectangular, but we’re only going to need square matrices • E.g. Wait…that doesn’t tell us how to compute the cross product!

5. Math 200 Definition for 2x2 matrices: Determinants • E.g.

6. Math 200 Definition for 3x3: • Defined in terms of 2x2 determinants • Take away the row and column and take the 2x2 determinant

7. Math 200 Defining the Cross product

8. Math 200 Let’s compute the cross product of the vectors v and w. Example

9. How do we determine if our answer is correct? Orthogonal to the original two vectors (right-hand rule) Dot product is 0! Math 200 Checking our work

10. Math 200 Write out a copy of the matrix next to the original. Diagonals going right are positive, to the left are negative Another way to think about computing the cross product

11. Math 200 The norm of the cross-product is the area of the parallelogram formed by the two vectors. w Area v • This also means that

12. Math 200 Compute the cross-product of the vectors v = <-2,1,1> and w = <3,1,2> Check your answer by computing the dot product of your answer with each of the vectors v and w Plot the three vectors on Geogebra 3D • https://www.geogebra.org/ Compute the area of the parallelogram formed by v and w Example

13. Math 200 v x w = = < 2 - 1, 3 - (-4), -2 - 3 > = <1, 7, -5> • <1, 7, -5> • < -2, 1, 1 > = -2 + 7 - 5 = 0 • <1, 7, -5> • < 3, 1, 2 > = 3 + 7 - 10 = 0 • ||< 1, 7, -5 >|| = (12 + 72 + 52)1/2 =

14. Definition: the scalar triple product is a • (v x w) Because it’s a dot product, we’ll get a scalar at the end It’s absolute value is the volume of a the parallelepiped formed by the three vectors: a Math 200 w One last property v

15. w C Math 200 Compute the area of a triangle given the three points… extra application A v B • Draw two vectors • Find their cross product v x w • Compute ||v x w||/2 (divided by two because the triangle is half of the parallelogram)