6.1 - Cross Product and Vol of Parallelipiped Flashcards
1
Q
Def: Cross product of vectors in R3 using i, j, and k.
A
(a1, a2, a3) x (b1, b2, b3)
det( i j k )
a1 a2 a3
b1 b2 b3
Solve using matrix cofactos to get
det( a2 a3, b2, b2) - det(a1 a3, b1 b3) + det(a1 a2, b1 b2)
2
Q
Theory: Relate axb and vector a and b
A
axb is orthogonal to both a and b.
3
Q
Theory: 4 properties of cross product.
A
- axb is perp to both a and b
- axb = -(bxa)
- norm(axb) = norm(a)norm(b)sint. where t is btwn a and b
- k(axb) = axkb = kaxb
4
Q
Def: Area of parallelogram.
A
Cross product between vectors composing both sides.
5
Q
Theory: Cross product in 2-D.
A
a = (x y) becomes a* = (x y 0)
6
Q
Def: Triple Scalar product.
A
c * axb
7
Q
Theory: Volume of parallelepiped.
A
absolute value of triple scalar product.
8
Q
Theory: Notation of absolute value vs norm.
A
abs = | x |
norm = || x ||
