Cross product

Cross product

Lessons

Notes:

Cross Product
Suppose given two vectors AA and BB, you want to find a third vector that is perpendicular to them. To find the third vector, we need to do the cross product.
Let A=<a1,a2,a3>A= \lt a_1,a_2,a_3 \gt and B=<b1,b2,b3>B= \lt b_1,b_2,b_3 \gt . Then the cross product of these two vectors will be:

A×B=<a2b3b2a3,b1a3a1b3,a1b2a2b1>A \times B = \lt a_2 b_3 - b_2 a_3 , b_1a_3 -a_1b_3 , a_1 b_2 - a_2 b_1 \gt

The formula is ugly to remember, so we have a technique that we will show you in the introduction videos!

Cross Product is NOT commutative
Keep in mind that

A×BB×AA \times B \ne B \times A

However, the relationship between these two cross products is that they are pointing in different directions. Hence,

A×B=(B×A)A \times B = -(B \times A)

Properties of Cross Products
Let u,vu,v and ww be vectors and cc is a scalar. Then we have the following cross product properties:
  1. u×(v+w)=u×v+u×wu \times (v+w) = u \times v + u \times w
  2. (u+v)×w=u×w+v×w(u + v) \times w = u \times w + v \times w
  3. (cu)×v=c(u×v)=u×(cv) (cu) \times v = c(u \times v) = u \times (cv)
  4. u(v×w)=(u×v)w u \cdot (v \times w ) = (u \times v) \cdot w
  5. u×(v×w)=(uw)v(uv)w u \times (v \times w ) = (u \cdot w) v - (u \cdot v) w
  • Introduction
    Cross Product Overview
    a)
    Cross Product
    • A vector perpendicular to the other two
    • Formula for the cross product
    • Technique to use cross product

    b)
    Cross Product is Not Commutative
    • Order in which you do cross product matters!
    • A×BB×AA \times B \ne B \times A
    • A×B=(B×A)A \times B = -(B \times A)

    c)
    Properties of Cross Product
    • Algebraic properties
    • Distribution property
    • Scalar multiplication property
    • Etc.


  • 1.
    Using the Cross Product
    If u=<3,2,4>u=<3, -2, 4> and v=<1,2,5>v=<-1, 2, -5>, then compute u×vu \times v.

  • 2.
    If u=<0,1,3>u= <0, 1, -3> and v=<1,0,4>v=<1, 0, 4>, then compute u×vu \times v.

  • 3.
    Showing Cross Product is Not Commutative Let A=<1,0,3>A= <1, 0, 3> and B=<2,1,0>B= <-2, 1, 0>. Show that:

    A×BB×AA \times B \ne B \times A


  • 4.
    Verifying Properties of Cross Product Suppose c=2c=2, and u=<2,1,4>u=<2, -1, 4> and v=<4,2,1>v=<4, 2, 1>. Verify that

    (cu)×v=c(u×v) (cu) \times v = c(u \times v)


  • 5.
    Suppose u=<2,1,4>u=<2, -1, 4>, v=<4,2,1>v=<4, 2, 1> and w=<1,1,2>w=<1, -1, 2>. Verify that

    u×(v+w)=u×v+u×wu \times (v + w) = u \times v + u \times w