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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 threedimensional 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 righthanded arrow over the letter to denote a vector. An ndimensional vector A has n elements denoted as A1, A2, ..., An. Symbolically, this can be written in multiple ways:
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
Geometrically speaking, magnitude is synonymous with "length," "distance", or "speed." In the twodimensional 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 threedimension 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 threedimensional 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
Note: Addition of vectors is only defined if both vectors have the same dimension. Example:
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
Example:
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. 


