Chapter 3.2 Flashcards
If v=(v(1), v(2), … , v(n)) is a vector in R^(n), what is the norm of v?
||v|| = sqrt(v(1)^2, v(2)^2, … , v(n)^2)
||v|| can be called three things?
1) Norm of v (common mathematical synonym for length)
2) Length of v
3) Magnitude of v
If v is a vector in R^(n) and k is any scalar, then (3) (norm)
1) ||v|| => 0
2) ||v|| = 0 iff v=0
3) ||kv|| = |k|||v||
Unit vectors
A vector of norm 1. You can obtain a unit vector with the same direction as any vector v by the formula
u = 1/||v||*v This is called “normalizing v”
Standard unit vectors
Unit vectors in positive directions of coordinate axes
If the standard unit vectors in R^(n) are e1, e2, … en, then any vector v in R^(n) can be expressed as
v = (v(1)e1, v(2)e2, … , v(n)*en)
If v = (v(1), v(2), … , v(n)) and u = (u(1), u(2), … , u(n)) then we denote the distance between u and v to be
d(u,v) = sqrt((v(1)-u(1))^2, (v(2)-u(2))^2, … , (v(n)-u(n))^2)
The angle v between u and w satisfies
0 <= pi
If u and w are nonzero vectors in R^(n), and if v is the angle between u and w, then the dot product (also called?) of u and w is
u·w = ||u||||w||cos(v)
Also called Euclidian Inner Product
If u=0 or w=0 we define u·w to be 0
How is the angle v expressed through the dot product and what can be said about the angle through examination of the dot product
cos(v) = u·w/(||u||*||v||)
1) v is obtuse if u·w < 0
2) v is acute if u·w > 0
3) v is pi/2 uf u·w = 0
If v = (v(1), v(2), … , v(n)) and u = (u(1), u(2), … , u(n)) are vectors in R^(n) then the dot product (also called?) of u and v is denoted u·v and is defined by
u·v = (u(1)v(1), u(2)v(2), … , u(n)*v(n))
Also called the Euclidian Inner Product
If u, v and w are vectors in R^(n), what correlations between the dot product and the norm is there (5)
1) ||v||=sqrt(v·v)
2) v·v = (||v||)²
3) (u·v) = (||u+v||)²
4) |u·v| <= ||u||||v||
if u and v are codirectional:
5) u·v = ||u||||v||
If u, v and w are vectors in R^(n) and k and m are scalars then (5) (dot product)
1) u·v = v·u
2) u·(v+w) = u·v + u·w
3) k(u·v) = (ku)·v
4) u·u >= 0 and u·u = o iff u = o
5) 0·v = v·0 = 0
Cauchy-Schwartz Inequality
If v = (v(1), v(2), … , v(n)) and u = (u(1), u(2), … , u(n)) are vectors in R^(n) then
|u·v| <= ||u||*||v||
Triangle Inequalities for vectors and distances
1) ||u + v|| <= d(u,v) + d(v,w) (for distances)
