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Mathematics

MATH 115

Martin Pei

Fall

Description

MATH 115 - Linear Algebra for Engineers
Kevin Carruthers
Fall 2012
Vectors
Vectors have a magnitude and a direction, are denoted by vector arrows, and are said to
n
be in R
Two Main Operations
You can perform two main operations with vectors: vector addition, which is the algebraic
n
tail to tip addition of vectors, and scalar multiplication, whv where t R andv R
Linear Combinations
The linear combinatio of a set of vect~0;:::~nis any vector which can be obtained from
these vectors through vector addition and scalar multiplication. It has the ~ ++v
0 0
an nwhere a 0:::a n R
Example: for a = 1 and b = 0
0 1
3
3a 2b =
2
2
Any vector in R is a linear combination of this set.
1 Dot Product
2 3 2 3
a0 b0
In R , the dot product ou = 4::: and v =4 ::: is a scalar dened to be
an bn
~
a b = 0 0 + +n nb
thus ia = 3 and v = 2
5 6
u v = 3 2 + 5 6 = 36
Properties
u v = v u~
tu v = tu ~v), for any scalar t
u v + ~) = u v + u ~w
Magnitude
The magnitude of a vector is its length.
2 3
1
6 7
6 7
Example: for v =6 7
4 1
2 p
v = 3 + 4 = 5
More generally p
v = v v
and
v2= v v
Unit Vector
The unit vector is a vector of length one. Giv, the unit vector with the same direction
is
~v
2 This is called normalization.
Distance Between Points
For P and Q, the distance between them is PQ
Angle Between Two Vectors
For u and v, the angle between them is .
2 2 2
Deriving from the cosine law c = a + b 2abcos, we get
u ~v = u + ~v 2 ~uv cos
= (u ~v) u ~v)
= u u u ~v v u +~v v
= u 2u v + ~v2
2 ~uv cos = 2~u v
cos =
uv
1 u ~v
= cos
u v
Denition: two vectors u and v are orthogonal iu ~v = 0
Lines
In R ;y = mx + b, but in R we must two vectors to represent a line. The equation for a
n 2
line in R follows the same forms as in R , but uses any vector on the line as its intercept,
and any vector parallel to the line as its slope.
Example:
x = (3;1;4;1) + t(2;0;3;2);t R
3 Parametric Form
In parametric form, we solve for the values of each variable in the resulting vector. For
the above equation we have
8 x = 3 + 2t
> 0
< x 1 1
x =
> x 2 4 + 3t
:
x 3 1 + 2t
where t R
3
Planes (In R only)
3
Every plane in R has a normal vector orthogonal to the plane.n must be orthogonal to
PX, where P and X are both points on the plane, so
n PX = 0
2 3
a
For any line in R of the form ax + by + cz = n =~4b5
c
Projections
proj (v) is the projection ov ontou~
u
1. prou v) is a scalar multipleuof ~
2. prou v) andv proju(v) are orthogonal
u v
projuv) = 2u
u
perp uv) = v projuv)
perpu(v) prouv) = 0
Shortest Distance
The tip of prouv) is the closest point tv ~
4

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