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Physics Study NotesINTRODUCTION TO VECTORS
Vectors are geometric representations of magnitude and direction and
can be expressed as arrows in two or three dimensions. 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.
Vectors are geometric representations of magnitude and direction which
are often represented by straight arrows, starting at one point on a
coordinate axis and ending at a different point. All vectors have a length,
called the magnitude, which represents some quality of interest so that
the vector may be compared to another vector. Vectors, being arrows,
also have a direction. This differentiates them from scalars, which are
mere numbers without a direction.
A vector is defined by its magnitude and its orientation with respect to a
set of coordinates. It is often useful in analyzing vectors to break them
into their component parts. For two-dimensional vectors, these
components are horizontal and vertical. For three dimensional vectors,
the magnitude component is the same, but the direction component is
expressed in terms of xx, yy and zz.
, 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 three-dimensional 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.
Decomposing a Vector
To visualize the process of decomposing a vector into its components,
begin by drawing the vector from the origin of a set of coordinates. Next,
draw a straight line from the origin along the x-axis until the line is even
with the tip of the original vector. This is the horizontal component of the
vector. To find the vertical component, draw a line straight up from the
end of the horizontal vector until you reach the tip of the original vector.
You should find you have a right triangle such that the original vector is
the hypotenuse.
Decomposing a vector into horizontal and vertical components is a very
useful technique in understanding physics problems. Whenever you see
motion at an angle, you should think of it as moving horizontally and
vertically at the same time. Simplifying vectors in this way can speed
calculations and help to keep track of the motion of objects.
Scalars and Vectors: Mr. Andersen explains the differences between
scalar and vectors quantities. He also uses a demonstration to show the
importance of vectors and vector addition.
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