MATH 200 WEEK 2 MONDAY CROSS MATH 200

MATH 200 WEEK 2 - MONDAY CROSS

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

MATH 200 WAIT…THAT DOESN’T TELL US HOW TO COMPUTE THE CROSS ▸ True, but before doing that we PRODUCT! 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 2 x 2 matrices: ▸ E. g.

MATH 200 ▸ Definition for 3 x 3: ▸ Defined in terms of 2 x 2 determinants ▸ Take away the row and column and take the 2 x 2 determinant

MATH 200 DEFINING THE CROSS PRODUCT

MATH 200 EXAMPLE ▸ Let’s compute the cross product of the vectors v and w.

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!

MATH 200 ANOTHER WAY TO THINK ABOUT COMPUTING CROSS ▸ Write out a copy of. THE the matrix next to PRODUCT 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 3 D ▸ https: //www. geogebra. org/ ▸ Compute the area of the parallelogram formed by v and 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: a w v

MATH 200 EXTRA APPLICATION ▸ Compute the area of a triangle given the three points… w C 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)