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Vectors (introduction to vectors)
Terms
Direction - The direction in which 2D-vector points can be characterized by
a single angle; for 3D-vectors two angles are needed.
Euclidean Space - The name given to all finite-dimensional spaces obtained
by taking Cartesian products of the real numbers R . They are denoted by R n
for n = 1, 2, 3,
Magnitude - The magnitude of a vector is its length, or distance from the
origin.
Projection - The projection of a vector in a particular direction is its
"shadow" along that direction. If u is a unit vector, the projection of a vector
v in the direction of u is given by a new vector which points in the direction
of u and whose magnitude is vƒu: i.e. the projection of v in the direction of u
is precisely (vƒu) u.
Right-hand-rule - This is the standard convention chosen when defining the
cross product between two vectors. It states that i×j = k, instead of -k, even
though both options are equally valid. Once this convention has been
chosen, there is no longer any ambiguity about whether the cross product
between two vectors points upwards or downwards. (Before this we only
knew it had to point in a direction perpendicular to the plane of the original
two vectors).
Rotational invariance - A vector quantity (such as the dot product or the
cross product) is rotationally invariant if its value remains the same under a
rotation of its input vectors. Both the dot product and the cross product are
rotationally invariant, while vector addition and scalar multiplication, in
general, are not.
Scalar - An ordinary number; whereas vectors have direction and
magnitude, scalars have only magnitude. The scalars we will be dealing with
will all be real numbers, but other kinds of numbers can also be scalars. 5
miles represents a scalar.
Unit vector - A vector whose length is one. The unit vectors which point in
the x -, y -, and z -directions in typical 3-dimensional space are usually
denoted by i , j , and k , respectively.
Vector - A two-dimensional vector is an ordered pair (a, b) of numbers; a
three-dimensional vector is an ordered triplet (a, b, c) . In other words,
points in the plane or in three-dimensional space are vectors. These kinds of
vectors can also be described as having direction and magnitude: 5 miles to
the east represents a vector.
Physics Notes By Xitclub.com
Vectors (introduction to vectors)
Vector Space - A set that is closed under addition and scalar multiplication.
Examples of vector spaces include the Euclidean plane R 2 and ordinary
three- dimensional space R 3
Physics Notes By Xitclub.com
Vectors (introduction to vectors)
Introduction to Vectors
In order to represent physical quantities such as position and momentum in more than one
dimension, we must introduce new mathematical objects called vectors. Technically
speaking, a vector is defined as an element of a vector space, but since we will only be
dealing with very special types of vector spaces (namely, two- and three-dimensional
Euclidean space) we can be more specific. For our purposes, a vector is either an ordered
pair or triplet of numbers. On a two-dimensional plane, for instance, any point (a, b) is a
vector. Graphically, we often represent such a vector by drawing an arrow from the origin
to the point, with the tip of the arrow resting at the point. The situation for threedimensional vectors is very much the same, with an ordered triplet (a, b, c) being
represented by an arrow from the origin to the corresponding point in three-dimensional
space.
The vector (a, b) in the Euclidean plane.
Unlike scalars, which have only a value for magnitude, vectors are often described as
objects that have both magnitude and direction. This can be seen intuitively from the
arrow-like representation of a vector in the plane. The magnitude of the vector is simply
the length of the arrow (i.e. the distance from the point to the origin), and can be easily
computed using the Pythagorean Theorem. The direction of a vector in two dimensions
can be characterized by a single angle θ (see ); the direction of a vector in three
dimensions can be specified using two angles (usually denoted θ and μ ).
While these ideas are perfectly valid in our case (since we're dealing with vectors in
finite-dimensional Euclidean space) it is not a good idea to become too attached to the
notions of "direction" and "magnitude" for vectors. For instance, in quantum mechanics
vectors often come in the form of functions (for instance, a particle wave function), and
in such a case it doesn't make sense to talk about the "direction" of the vector. We don't
have to worry about these complications for now
Physics Notes By Xitclub.com