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(6) = 2.449..., which means that the person is traveling at about 2.449 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. | |
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