1. Vectors →
The vector from A to B is denoted by AB; the point A is called its initial point, or
tail, and the point B is called its terminal point, or head. The individual
coordinates are called the components of the vector.
u + v = [u1 + v1, u2 + v2] vector addition
cv = c[v1, v2] = [cv1, cv2] scalar multiplication
Let u, v, and w be vectors in R n
and let c and d be scalars. Then
a) u+v=v+u
b) (u + v) + w = u + (v + w)
c) u+0=u
d) u + (– u) = 0
e) c(u + v) = cu = cv
f) (c + d)u = cu + du
g) c(du) = (cd)u
h) 1u = u
A vector v is a linear combination of vectors v1, v2, …, vk if there are scalars
c1, c2, …., ck such that u
v = c1v1 + c2v2 + …. + ckvk. The scalars c1, c2, …, ck are called the coefficients of
the linear combination.
[] []
u1 v1
u2 v2
If u = . and v = . then the dot product u · v of u and v is defined by
. .
un vn
u · v = u1v1 + u2v2 + … + unvn
Let u, v, and w be vectors in R n
and let c be a scalar. Then
a) u · v = v · u
b) u · (v + w) = u · v + u · w
c) (cu) · v = c(u · v)
d) u · u ≥ 0 and u · u = 0 if and only if u = 0
[]
v1
v2
The length of a vector v = . in R n
is the nonnegative scalar ‖v‖
.
vn
defined by
‖v‖ = √ v ∙ v = √ v +v +…+ v
2
1
2
2
2
n
,Let v be a vector in and let c be a scalar. Then
a) ‖v‖ = 0 if and only if v = 0
b) ‖c v‖ = |c|‖v‖
1
A vector of length 1 is called a unit vector. Normalizing a vector: v
‖v‖
The Cauchy-Schwarz Inequality
For all vectors u and v in R n
,
|u ∙ v|≤ ‖u‖‖v‖
, The Triangle Inequality
For all vectors u and v in R n
,
‖u+ v‖≤ ‖u‖+‖v‖
The distance d(u, v) between vectors u and v in R n
is defined by
d(u, v) = ‖u−v‖
For nonzero vectors u and v in R n
,
u∙ v
cos θ = ‖u‖‖v‖
Two vectors u and v in R n
are orthogonal to each other if u · v = 0.
Pythagoras’ Theorem
For all vectors u and v in R n
, ‖u+ v‖2 = ‖u‖2 + ‖v‖2 if and only if u and
v are orthogonal.
If u and v are vectors in R n
and u ≠ 0, then the projection of v onto u is
the vector proju(v) defined by
Proju(v) = ( u∙u ∙uv )u
Normal, general and vector form of a line l
The normal form of the equation of a line l in R 2
is
n · (x – p) = 0 or n·x=n·p
where p is a specific point on l and n ≠ 0 is a normal vector for l.
The general form of the equation of l if ax + by = c, where n = [ ab] is a
normal vector for l.
The vector form of the equation of a line l in R 2
or R 3
is x = p + td
where p is a specific point on l and d ≠ 0 is a direction vector for l.
The equations corresponding to the components of the vector form of the
equation are called
parametric equations of l.
Normal, general and vector form of a plane ℘
The normal form of the equation of a plane ℘ in R 3
is
The vector from A to B is denoted by AB; the point A is called its initial point, or
tail, and the point B is called its terminal point, or head. The individual
coordinates are called the components of the vector.
u + v = [u1 + v1, u2 + v2] vector addition
cv = c[v1, v2] = [cv1, cv2] scalar multiplication
Let u, v, and w be vectors in R n
and let c and d be scalars. Then
a) u+v=v+u
b) (u + v) + w = u + (v + w)
c) u+0=u
d) u + (– u) = 0
e) c(u + v) = cu = cv
f) (c + d)u = cu + du
g) c(du) = (cd)u
h) 1u = u
A vector v is a linear combination of vectors v1, v2, …, vk if there are scalars
c1, c2, …., ck such that u
v = c1v1 + c2v2 + …. + ckvk. The scalars c1, c2, …, ck are called the coefficients of
the linear combination.
[] []
u1 v1
u2 v2
If u = . and v = . then the dot product u · v of u and v is defined by
. .
un vn
u · v = u1v1 + u2v2 + … + unvn
Let u, v, and w be vectors in R n
and let c be a scalar. Then
a) u · v = v · u
b) u · (v + w) = u · v + u · w
c) (cu) · v = c(u · v)
d) u · u ≥ 0 and u · u = 0 if and only if u = 0
[]
v1
v2
The length of a vector v = . in R n
is the nonnegative scalar ‖v‖
.
vn
defined by
‖v‖ = √ v ∙ v = √ v +v +…+ v
2
1
2
2
2
n
,Let v be a vector in and let c be a scalar. Then
a) ‖v‖ = 0 if and only if v = 0
b) ‖c v‖ = |c|‖v‖
1
A vector of length 1 is called a unit vector. Normalizing a vector: v
‖v‖
The Cauchy-Schwarz Inequality
For all vectors u and v in R n
,
|u ∙ v|≤ ‖u‖‖v‖
, The Triangle Inequality
For all vectors u and v in R n
,
‖u+ v‖≤ ‖u‖+‖v‖
The distance d(u, v) between vectors u and v in R n
is defined by
d(u, v) = ‖u−v‖
For nonzero vectors u and v in R n
,
u∙ v
cos θ = ‖u‖‖v‖
Two vectors u and v in R n
are orthogonal to each other if u · v = 0.
Pythagoras’ Theorem
For all vectors u and v in R n
, ‖u+ v‖2 = ‖u‖2 + ‖v‖2 if and only if u and
v are orthogonal.
If u and v are vectors in R n
and u ≠ 0, then the projection of v onto u is
the vector proju(v) defined by
Proju(v) = ( u∙u ∙uv )u
Normal, general and vector form of a line l
The normal form of the equation of a line l in R 2
is
n · (x – p) = 0 or n·x=n·p
where p is a specific point on l and n ≠ 0 is a normal vector for l.
The general form of the equation of l if ax + by = c, where n = [ ab] is a
normal vector for l.
The vector form of the equation of a line l in R 2
or R 3
is x = p + td
where p is a specific point on l and d ≠ 0 is a direction vector for l.
The equations corresponding to the components of the vector form of the
equation are called
parametric equations of l.
Normal, general and vector form of a plane ℘
The normal form of the equation of a plane ℘ in R 3
is
