Math 200 Week 2 - Monday Cross Product

Main Questions for Today Math 200 Main Questions for Today 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?

Math 200 Definition 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

Wait…that doesn’t tell us how to compute the cross product! Math 200 Wait…that doesn’t tell us how to compute the cross product! 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.

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

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

Defining the Cross product Math 200 Defining the Cross product

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

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

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

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

Math 200 Example 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

<1, 7, -5> • < -2, 1, 1 > = -2 + 7 - 5 = 0 Math 200 i j k -2 1 3 2 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 =

One last property Definition: the scalar triple product is a • (v x w) Math 200 One last property 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: v w a

Math 200 extra application Compute the area of a triangle given the three points… A B C v w 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)