Chapter 3.5 Flashcards
Def: The cross product of u×v (if u=(u1,u2,u3) and v=(v1,v2,v3) are vectors in 3-space)
u×v = (u2v3-u3v2, u3v1-u1v3, u1v2-u2v1)
Trick: u×v from a 2×3 matrix with u as first row and v as second?
The first component is the determinant of the submatrix obtained by crossing out the first column.
The second component is the negative of the determinant of the submatrix obtained by crossing out the second row.
The third component is the determinant of the submatrix obtained by crossing out the last column.
If u, v, and w are vectors in R³ and k is any scalar, then (10)
(4 dot product) (4 arithmatic rules) (2 zero properties)
1) u·(v×u) = v·(u×v) = u·(u×v) = v·(v×u) = 0
2) ||u×v||² = ||u||²||v||² - (u·v)² (Laranges identity)
3) u×(w×v) = (u·w)v-(u·v)w
4) (u×v)×w = (u×w)v - (v×w)u
5) u×v = -(v×u)
6) u×(v+w) = u×v + u×w
7) (u+v)×w = u×w + v×w
8) k(u×v) = (ku)×w = u×(kv)
9) u×0 = 0×u = 0
10) u×u = 0
In what symbolic form can the cross product be represented?
u×v can be obtained if we construct the 3×3 matrix A with the row vectors r1 = (i, j, k), r2 = u and r3 = v.
u×v = (C(r1), C(r2), C(r3) )
What two examples show that it is not generally true that u×(v×w) = (u×v)×w?
i×(j×j) = i×0 = 0 (i×j)×j = k×j = -i
Explain the “right hand rule”
If you curve your fingers through u to v, your thumb will point in the direction of u×v
Larange’s identity?
||u×v||² = ||u||²||v||² - (u·v)²
What is the norm of u×v?
||u×v|| = ||u||||v||sin(v)
Area of a parallellogram in 3-space?
If u and v are vectors in 3-space then ||u×v|| is equal to the area of the parallellogram determined by u and v
What is the scalar triple product?
u·(v×w) if u, v and w are vectors in 3-space
It can be calculated as the determinant of the 3×3 matrix A constructed with the row vectors r1=u r2=v r3=w
Why doesn’t the formula (u·v)×w make sense?
Because you cannot take the cross product of a vector and a scalar.
What correlation between the determinant and space exists?
The absolute value of the determinant of a square matrix is equal to the volume* of the space the row vectors spann. If two vectors are linearly dependent the volume is zero.
|u·(v×w)| = ?
The volume of the parallellpiped determined by u, v and w.
Also expressed u·(v×w) = +-V where +- depend on whether the angle between u and v×w is acte (positive) or obtuse (negative) or zero (they’re all coplanar.
Vectors u, v and w in 3-space (with common initial point) lie in the same plane iff?
u·(v×w) = 0
