1. Math 200
Week 2 - Monday
Cross Product

2. 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?

3. 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

4. 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.

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

6. 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

7. Defining the Cross product
Math 200
Defining the Cross product

8. 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.

9. 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!

10. 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

11. 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

12. 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
Compute the area of the parallelogram formed by v and w

13. <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), > = <1, 7, -5>
<1, 7, -5> • < -2, 1, 1 > = = 0
<1, 7, -5> • < 3, 1, 2 > = = 0
||< 1, 7, -5 >|| = ( )1/2 =

14. 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

15. 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)