Introduction
Physical quantities described by just a number: scalars
Physical quantities described by a number and a direction: vectors
More generally, vectors can represent a collection of ordered numbers
v=(v1,v2,…,vn)
Introduction
Vectors are very useful in physics to deal with concepts such as force, speed, fields, etc.
They are also useful to the treatment of data in general
Example: the cosine of the angle between two vectors is often used in text analysis to determine the similarity between documents
Basic Operations
A number of basic operations can be applied to vectors:
→ Addition
→ Subtraction
→ Multiplication by a scalar
Vector Operations
Additionaly, we can define:
→ The norm of a vector
→ The angle between two vectors
→ The dot product of two vectors
Addition
v+w=(v1+w1,v2+w2,…,vn+wn)
Subtraction
w−v=w+(−v)=(w1−v1,w2−v2,…,wn−vn)
Multiplication by a Scalar
kv=(kv1,kv2,…,kvn)
Properties
Basic Operations on vectors follow these properties:
→ Commutative
→ Associative
→ Distributive (over a scalar)
→ Identities (scalar and vector)
Norm
The norm of a vector is a measure of its length
It is represented by:
∣∣v∣∣
And, it can calculated as:
∣∣v∣∣=v12+v22+…+vn2
Norm properties
The norm of a vector follows these properties:
∣∣v∣∣≥0
∣∣v∣∣=0 if and only if v=0
∣∣kv∣∣=∣k∣∣∣v∣∣
Normalizing Vectors
A vector of norm 1 is called a unit vector
Vectors can be normalized:
u=∣∣v∣∣v
A vector with a 1 in one position, and zeroes elsewhere is a standard unit vector
Any vector can be written as a linear combination of standard unit vectors
Distance
The distance between two vectors is defined as
d(u,v)=∣∣u−v∣∣
=(u1−v1)2+(u2−v2)2+…+(un−vn)2
Dot Product
The dot product of vectors u and v is defined as:
u⋅v=∣∣u∣∣∣∣v∣∣cosθ
Therefore, cos(θ), can be calculated as:
∣∣u∣∣∣∣v∣∣u⋅v
Dot Product
Using the law of cosines, one can show that:
u⋅v=0.5(∣∣u∣∣2+∣∣v∣∣2−∣∣u−v∣∣2)
u⋅v=u1v1+u2v2+…+unvn
Dot Product Properties
The dot product follows these properties:
u⋅v=v⋅u
u⋅(v+w)=v⋅u+v⋅w
k(u⋅v)=(ku)⋅v
v⋅v≥0 , and v⋅v=0 if and only if v=0
Vectors and Linear Systems
Introduction
Previously, we showed that two points represent a line in a plane
One can also show that a point and a vector also represent a line
x=x0+kv
Introduction
The same way, a point and two vectors represent a plane in a 3D space
x=x0+k1v1+k2v2
Parametric Equations
Given an equation of the type:
ax+by+cz=d
One can find its vector parametric equations as:
x=a1(d−bt1−ct2), y=t1, z=t2
Parametric Equations
So, we have:
(x,y,z)=(a1(d−bt1−ct2),a1t1,a1t2)
Or, conversely:
(x,y,z)=(ad,0,0)+t1(−ab,a1,0)+t2(−ac,0,a1)
Dot Product of Linear Systems
A linear equation such as:
a1x1+a2x2+…+anxn=b
Can be rewriten in dot product as:
x⋅a=b,
where x=(x1,x2,…,xn) and a=(a1,a2,…,an)
Dot Product of Linear Systems
For a homogeneous system of equations:
a11x1a21x1⋮am1x1+a12x2+a22x2+⋮+am2x2+⋯++⋯++⋯++⋯+a1nxna2nxn⋮amnxn=0=0=⋮=0
Dot Product of Linear Systems
We can rewrite it as:
r1⋅xr2⋅x⋮rm⋅x=0=0=⋮=0
Dot Product of Linear Systems
This means that every possible solution x is orthogonal to each vector r
Example
112202−1−10x0x1x2=000
Solution
(x0,x1,x2)=t(1,0,0)−t(0,1,0)+t(0,0,1)
Dot Product of Linear Systems
The solution of each non-homogeneous system is equal to that of a homogeneous systems plus a fixed vector
Example
112202−1−10x0x1x2=111
Solution
(x0,x1,x2)=(1,0,0)+t(1,0,0)−t(0,1,0)+t(0,0,1)
Suggested Reading
Textbook: Sections 3.1, 3.2, 3.3, and 3.4
Exercises
→ Section 3.1: 1, 2, 9, 10
→ Section 3.2: 1, 2, 3, 4, 9, 10, 17, 18
→ Section 3.3: 1, 2
→ Section 3.4: 20, 25, 26, 27