Chapter 3: Euclidean vector spaces Flashcards
||kv||
= |k| ||v||
Angle between 2 vectors using cos
cosø = <strong>u•v</strong>/||<strong>u</strong>|| ||<strong>v</strong>||
Cauchy-Schwarz Inequality
|u•v| ≤ ||u|| ||v||
Parallelogram Equation for Vectors
||u + v||2 + ||u - v||2 = 2(||u||2 + ||v||2)
Euclidian inner product:
u•v = 1/4||u + **v||2 - 1/4 ||u** - v||2
Link between dot-product multiplication by matrix A and A transpose:
*A *u • **v **= **u • **AT v
u • Av = AT u • v
A normal to a line
ax + by + c = 0
**n **= (a, b)
Homogeneous equations in 2 or 3 unknowns can be written in vector form:
**n • x ** = 0
where n is the vector of coefficients
x is the vector of unknowns
proj<strong>a</strong>**v **
( v•u )u
where u = <strong>a</strong>/||a||, the unit vector of a.
If **u **and v are orthogonal vectors in Rn with the Euclidean inner product, then
||u + v||2 = ||u||2 + ||v||2
Distance between the Point P(x0, y0) and the line ax + by + c = 0
D =
||proj<strong>a</strong>u||
=|<strong>u•a</strong>|/||<strong>a</strong>||
= ||<strong>u</strong>|| |cosø|
Line through x0, parallel to v
x = x0 + *t *v
Plane through x0, parallel to v1 and v2.
x = x0 + t1 v1 + t2 v2
Line segment from **x0 **to x1.
x = **x0 + t** **(x1** - x0)
( 0 ≤ t ≤ 1 )
All vectors in Rn that are orthogonal to every row vector of a matrix A.
Solution set to:
Ax = 0
The general solution of a consistent linear system Ax = b, using the general solution of Ax = 0.
Obtained by adding any specific solution of Ax = b to the general solution of Ax = 0.
4 Relationships involving Cross Product and Dot product:
- u • (** u** x v ) = 0 (u x v is orthogonal to u. (and to v))
- || u x v ||2 = ||u||2||v||2 - (u•v)2
- u x (** v** x w ) = ( u • **w )v** - ( u • **v )w**
- ( u x v ) x **w = ( u • **w )v - ( v • w )u
3 properties of the cross product
- u x v = - (v x u)
- k(u x v) = (ku) x v = u x (kv)
- u x u = 0
Angle between two vectors in terms of sin
sinø = ||<strong>u</strong>x <strong>v</strong>||/||<strong>u</strong>|| ||<strong>v</strong>||
|| u x v || in 3 space geometry, equals
the area of the parallelogram determined by u and v.
Scalar triple product of u, v, and w:
u • ( v x w )
( = w • ( u x v ) )
( = v • ( w x u ) )
The absolute value of the determinant:
Equals the area of the parallelogram in 2-space determined by the vector u = (u1, u2) and v = (v1, v2)
The absolute value of the determinant:
(in geometric terms)
Equals the volume the parallelepiped in 3-space determined by the vector u, v and w.
If the vectors u, v and w have the same initial point, then they ly in the same plane iff
The scalar triple product = 0
u • (v x w) = 0
