Chapter 13, Section 13.4: The Cross Product
for a 2x2 matrix of numbers (a b c d), det
( a b c d) = ad - bc
vt j x vt i =
-vt k (THINK RIGHT HAND RULE)
for any vt v, vt w, vt v x vt w =
-vt w x vt v
vt v x vt v =
0 (vectors are parallel)
find a vector parallel to the intersection of two planes
1. line of intersection will be a line in both planes 2. if vt n is normal to the 1st plane and vt n2 is normal to the 2nd plane, then L will be perpendicular to both vt n and vt n2 3. so line of intersection is parallel to line given by vt n x vt n2
right hand rule
1. place hand on x with fingers curling towards y 2. whatever way your thumb points out is z
matrix analysis
1. the first row is the first vector being crossed (i, j, k values) 2. the second row is the second vector being crossed with the first (i, j, k) values
how to find the equation of a plane through 3 points
1. use points to find 2 vectors parallel to the plane (displacement vectors are parallel) 2. take cross product of the 2 parallel vectors to find a normal vector 3. use normal vector-point equation of the plane
properties of the cross product
1. vt v x vt w = -vt w x vt v 2. (λvt v) x vt w = λ(vt v x vt w) = vt v x (λvt w) 3. unit vt u x (vt v + vt w) = (vt u x vt v) + (vt u x vt w)
the determinant of a 3x3 matrix is determined by
3, 2x2 determinants
define vt v x vt w =
A(vt n) = ||vt v|| ||vt w|| sin(theta) ex: where n = unit normal to the plane containing vt v and vt w, determined by right hand rule
vt v x vt w is
a normal vector to the plane, direction given by right-hand rule, length = area of parallelogram
any 2 vectors in 3-space form
a parallelogram
determinant
a square array of numbers or variables enclosed between two parallel lines
cross product gives you
a vector
||vt v x vt w||
area of parallelogram determined by vt v and vt w
vt v x vt w is perpendicular to
both vt v and vt w ex: vt v * (vt v x vt w) = 0 vt w * (vt v x vt w) = 0
determinant rule thingy
cover up column, cross and subtract per determinant, then subtract the first determinant, then add the second
vt v x vt w algebraically
determinants -- see 13.4A
vt a x (vt b * vt c)
no - cannot cross a vector with a scalar
tan =
sin/cos
tan is basically
slope
length of the cross product is
the area of the parallelogram
if vt v and vt w are not parallel,
then they determine a plane and a parallelogram of area A = ||vt v|| ||vt w|| sin(theta)
geometric definition of the cross product
two vectors determine a plane, in fact a parallelogram of determined area
vt i x vt j =
vt k (THINK RIGHT HAND RULE)
algebraic definition of the cross product: part 1
vt v = v1(vt i) + v2(vt j) + v3(vt k) vt w = w1(vt i) + w2(vt j) + w3(vt k)
if vt v and vt w are parallel,
vt v x vt w = 0
vt a * (vt b x vt c)
yes - cross product of two vectors ex: called TRIPLE PRODUCT (scalar)
triple product: |vt a * (vt b x vt c)| =
|det(a1 a2 a3 b1 b2 b3 c1 c2 c3)| = volume of 3d parallelogram spanned by vt a, vt b, and vt c
||vt v x vt w|| =
||vt v|| ||vt w|| sin(theta)