Vector,
quantity having both magnitude and direction. For example, an ordinary quantity,
or scalar, can be exemplified by the distance 6 km; a vector quantity can be
exemplified by the term 6 km north. Vectors are usually represented by directed
line segments, such as B in the diagram below; the
length of the line segment is a measure of the vector quantity, and its
direction is the same as that of the vector.
The simplest use of vectors and calculation by means of
vectors is illustrated in the diagram, drawn to represent a boat moving across a
stream. Vector a, or A, indicates the motion of
the boat in the course of a given interval of time if it were moving through
still water; vector b, or $, shows the drift or flow of
the current during the same period of time. The actual path of travel of the
boat under the influence of its own propulsion and of the current is represented
by vector c, or B. By the use of vectors any
type of problem involving the motion of an object being acted on by several
forces can be solved graphically.
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This method of problem solution, known as vector addition,
is performed as follows. A vector representing one force is drawn from the
origin O in the proper direction. The length of the vector is made to
agree with any convenient arbitrary scale, such as a given number of centimeters
to the kilometer. In the diagram the rate of rowing was 2.2 km/h, the time rowed
was 1 hr, and the scale is 1 cm to 1 km. Therefore, the line A is drawn as 2.2 cm to equal
2.2 km. The current speed of 6 km/h is then represented by a vector $ that is 6 cm long,
indicating a distance of 6 km that the current moved during 1 hr. This second
vector is drawn with its origin at the end of vector a in a direction
parallel to the flow of the current. The point B at the end of the second
vector represents the actual position of the boat at the end of 1 hr of travel,
and the actual distance traveled is represented by the length (in this case,
about 6.4 km) of the vector c, or B.
Problems in vector addition and subtraction such as the
one above can be easily solved by graphic methods and can also be calculated by
means of trigonometry. This type of calculation is useful in solving problems in
navigation and motion as well as in mechanics and other branches of physics. In
present-day advanced mathematics, a vector is considered an ordered set of
quantities with appropriate rules of manipulation. Vector analysis, that is, the
algebra, geometry, and calculus of vector quantities, enters into the applied
mathematics of every field of science and engineering.
