How are vectors added?
One way to combine vectors is through addition (or composition). This can be done
algebraically or graphically. For example, to add the two vectors U [3, 1] and V [5, 2],
one can add their corresponding components to find the resultant vector R [2, 3]. One
also can graph U and V on a set of coordinate axes, completing the parallelogram
formed with U and V as adjacent sides, obtaining R as the diagonal from the common
vertex of U and V.How is the productof two vectorsdetermined?
There are two distinct types of products of two vectors: scalar and vector products,
sometimes called the inner and outer products (mostly in reference to tensor prod-
ucts; see below). The scalar (or dot) product of two vectors is not a vector because the
product has a magnitude but not a direction. For example, if Aand Bare vectors (of
magnitude Aand B,respectively), their scalar product is: A• BABcos , in which
is the angle between the two vectors. This scalar quantity is also called the dot product
of the vectors. These equations obey the commutative and distributive laws of algebra
(for more information, see “Algebra”). Thus, A• BB• A; A• (BC) A• BA- C. If Ais perpendicular to B, then A• B0.
The vector (or cross or skew) product of Aand Bis the length CABsin ; its
direction is perpendicular to the plane determined by Aand B. In this case, this kind
of multiplication does not follow the commutative law, as A• BB• A.
What is vector analysis?
Vector analysis is the calculus of functions with variables as vectors—a part of calcu-
lus also known for its derivative and integral equations. The components of a vector
do not always have to be constants. They can also be variables and functions of vari-
ables, such as the position of a body moving through space represented by a vector
whose x, y,and zcomponents are all functions of time. In this case, the calculus can
be used to solve such vector functions, which are also called vector analysis.What is a tensor?
A tensor is a quantity that depends linearly on many vector variables. They are consid-
ered to be a set of n' components that are functions of the coordinates at any point in
ndimensional space. Tensors are used in several fields of mathematics, such as the
theory of elasticity (stress and strain) and mathematical physics, especially with
regard to the theory of relativity.What are some other typesof analysis?
There are numerous other analyses in mathematics and the sciences besides vector
240 analysis. At one time, the study of tensors was known as the absolute differential cal-