The Cross Product of 2 Vectors 11.3

1. The Cross Product of 2 Vectors11.3 JMerrill, 2010

2. Unit Vectors in 2D • In 2-D space, the unit vectors <0,1> and <1,0> are the standard unit vectors and denoted by i = <1,0> and j = <0,1> j = <0,1> i = <1,0>

3. Unit Vectors in 2D • Any vector can be written as a linear combination of the vectors I and j. • v = <v1, v2> = v1<1,0> + v2<0,1> = v1i + v2j • The scalars v1and v2 are the horizontal and vertical components of v.

4. Writing a Linear Combination of Unit Vectors • u has initial point (2, -5) and terminal point (-1,3). Write u as a linear combination of the unit vectors i & j. • u= <-1-2, 3-(-5)> = <-3, 8> = -3i + 8j

5. Unit Vectors in 3D

6. The Cross Product

7. Finding The Cross Product • An easy way to calculate the cross product is to use a matrix. We use the determinant form with cofactor expansion.

8. Finding The Cross Product • An easy way to calculate the cross product is to use a matrix. We use the determinant form with cofactor expansion.

9. Finding the Cross Product Subtraction sign Addition Sign

10. Example • Given u = i + 2j + k and v = 3i + j + 2k, find the cross product of u x v.

11. You Try • Given u = i + 2j + k and v = 3i + j + 2k, find the cross product of v x u.

12. Using the Cross Product • Find a unit vector that is orthogonal to both u = 3i – 4j + k and v = -3i + 6j. • The cross product gives a vector that is orthogonal to both u and v = -6i – 3j + 6k • The question asks for a unit vector that’s orthogonal.

13. Using the Cross Product • So, we need to divide by the magnitude of the orthogonal vector. • -6i – 3j + 6k

14. Triple Scalar Product • Given 3 vectors u = 3i – 5j + k v = 2j – 2k w = 3i + j + k • Find the volume of a parallelepiped having these vectors as adjacent edges. • The volume is found by V = |u∙(v x w)|

15. Triple Scalar Product