Prelude: A vector, as defined below, is a specific mathematical structure.
It has numerous physical and geometric applications, which result mainly
from its ability to represent magnitude and direction
simultaneously. Wind, for example, had both a speed and a direction and,
hence, is conveniently expressed as a vector. The same can be said of
moving objects and forces. The location of a points on a cartesian coordinate
plane is usually expressed as an ordered pair (x, y), which is a specific
example of a vector. Being a vector, (x, y) has a a certain distance (magnitude)
from and angle (direction) relative to the origin (0, 0). Vectors are
quite useful in simplifying problems from three-dimensional geometry.
Definition:A scalar,
generally speaking, is another name for "real number."
Definition:A vector
of dimension n is an ordered collection of n elements, which are called
components.
Notation: We often represent a vector by some letter, just as we use
a letter to denote a scalar (real number) in algebra. In typewritten work,
a vector is usually given a bold letter, such as A, to
distinguish it from a scalar quantity, such as A. In handwritten
work, writing bold letters is difficult, so we typically just place a
right-handed arrow over the letter to denote a vector. An n-dimensional
vector A has n elements denoted as A1, A2, ..., An. Symbolically,
this can be written in multiple ways:
A = <A1, A2, ..., An>
A = (A1, A2, ..., An)
Example: (2,-5), (-1, 0, 2), (4.5), and (PI, a, b, 2/3) are all examples
of vectors of dimension 2, 3, 1, and 4 respectively. The first vector
has components 2 and -5.
Note: Alternately, an "unordered" collection of n elements
{A1, A2, ..., An} is called a "set."
Definition: Two vectors are equal if their corresponding
components are equal.
Example: If A = (-2, 1) and B = (-2,
1), then A = B since -2 = -2 and 1 =
1. However, (5, 3) not_equal (3, 5) because even though they have the
same components, 3 and 5, the component do not occur in the same order.
Contrast this with sets, where {5, 3} = {3, 5}.
Definition: The magnitude of a vector A of dimension
n, denoted |A|, is defined as
|A| = sqrt(A1^2 + A2^2 + ... + An^2)
Geometrically speaking, magnitude is synonymous with "length,"
"distance", or "speed." In the two-dimensional case,
the point represented by the vector A = (A1, A2) has a distance from the
origin (0, 0) of sqrt(A1^2 + A2^2) according to the pythagorean theorem.
In the three-dimension case, the point represented by the vector A = (A1,
A2, A3) has a distance from the origin of sqrt(A1^2 + A2^2 + A3^2) according
to the three-dimensional form of the Pythagorean theorem (A box with sides
a, b, and c has a diagonal of length sqrt(a2+b2+c2) ). With vectors of
dimension n greater than three, our geometric intuition fails, but the
algebraic definition remains.
Definition: The sum of two vectors A
= (A1, A2, ..., An) and B = (B1, B2, ..., Bn) is defined
as
A + B = (A1 + B1, A2 + B2, ..., An + Bn)
Note: Addition of vectors is only defined if both vectors have the same
dimension.
Example:
(2, -3) + (0, 1) = (2+0, -3+1) = (2, -2).
(0.1, 2) + (-1, PI) = (0.1 + -1, 2 + PI) = (-0.9, 2+PI)
Justification: Physical and geometric applications warrant such a definition.
IF a train travels East at 5 meters/second relative to the ground, which
will be denoted in vector notation as VT = (0, 5), and a person on the
train walks South at 1 meter/second relative to the train, which will
be denoted as VP = (-1, 0), THEN the direction and speed that the person
is traveling relative to the ground is represented by the vector VG =
VT + VP = (0, 5) + (-1, 0) = (0 + -1, 5 + 0) = (-1, 5). This vector has
a magnitude of |VG| = sqrt((-1)^2 + 5^2) = sqrt(26) = 5.099..., which means
that the person is traveling at about 5.099 meters/second relative to
the ground and the net direction is mostly East but slightly South.
Definition: The scalar product of a scalar k by a vector
A = (A1, A2, ..., An) is defined as
kA = (kA1, kA2, ..., kAn)
Example:
2(5, -4) = (2*5, 2*-4) = (10, -8)
-3(1, 2) = (-3*1, -3*2) = (-3, -6)
0(3, 1) = (0*3, 0*1) = (0, 0)
1(2, 3) = (1*2, 1*3) = (2, 3)
Note: In general, 0A = (0, 0, ..., 0) and 1A
= A, just as in the algebra of scalars. The vector of
any dimension n with all zero elements (0, 0, ..., 0) is called the zero
vector and is denoted 0.
Orthogonal Vectors
To remember the right hand rule, write the
xyz order twice: xyzxyz. Next, find the pattern you’re looking for:
xy => z(xcrossyisz)yz => x(ycrosszisx; we looped around:ytoztox)zx => y
xy and yx have opposite signs because they are forward and backward in our xyzxyz setup.So, without a formula, you should be able to calculate:
Again, this is because
x cross y is positive z in a right-handed coordinate system. I used unit vectors, but we could scale the terms:
Example:
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