Representation of a vector
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Vectors are usually denoted in boldface, as a. Other conventions include a or a, especially in
handwriting. Alternately, some use a tilde (~) or a wavy underline drawn beneath the symbol, which
is a convention for indicating boldface type.
Vectors are usually shown in graphs or other diagrams as arrows,
called directed line segments. If the direction is from point A to B,
the directed line segment is denoted by AB or AB.
The point A is called the initial point, tail, base, start, or origin.
The point B is called the terminal point, head, tip, endpoint, or
destination.
The length of the arrow represents the vector's magnitude, while the direction in which the arrow
points represents the vector's direction.
The length or magnitude or norm of the vector a is denoted by |a|.
Any vector a can be written as a = axi + ayj with real numbers a1 and a2 which are uniquely
determined by a (i and j are the unit vectors parallel to the x-and y-axes respectively).
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Sometimes a is then also written as a pair of coordinates a  a x , a y  .
Length of a vector
The length of the vector a = axi + ayj in a three-dimensional Euclidean space can be computed with
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the Euclidean norm a  a x2  a y2 which is a consequence of the Pythagorean theorem.
Vector equality
Two vectors are said to be equal if they have the same magnitude and direction.
Vector addition and subtraction
Let a=axi + ayj and b=bxi + byj
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The sum of a and b is: a  b  a x  bx i  a y  b y  j
The addition may be represented graphically by placing the start of the arrow b at the tip of the
arrow a, and then drawing an arrow from the start of a to the tip of b. The new arrow drawn
represents the vector a + b, as illustrated below:
This addition method is sometimes called the parallelogram rule because a and b form the sides of
a parallelogram and a + b is one of the diagonals.
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
 
The difference of a and b is: a  b  a x  bx i  a y  b y  j
Subtraction of two vectors can be geometrically defined as follows: to subtract b from a, place the
ends of a and b at the same point, and then draw an arrow from the tip of b to the tip of a. That
arrow represents the vector a − b, as illustrated below:
Scalar multiplication
A vector may also be multiplied by a real number “r”. In mathematics, numbers are often called
scalars to distinguish them from vectors, and this operation is therefore called scalar
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multiplication. The resulting vector is: ra  ra x i  ra y  j
The length of ra is |r||a|. If the scalar is negative, it also changes the direction of the vector by 180o.
Two examples (r = -1 and r = 2) are given below:
Here it is important to check that the scalar multiplication is compatible with vector addition in the
following sense: r(a + b) = ra + rb for all vectors a and b and all scalars r. One can also show that a
- b = a + (-1)b.
Unit vector
A unit vector is any vector with a length of one. If you have a vector of arbitrary length, you can use
it to create a unit vector. This is known as normalizing a vector.
To normalize a vector a = [ax, ay], scale the vector by the reciprocal of its length ||a||. That is:
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ay
a
a
aˆ    x iˆ   ˆj
a
a
a
Dot product
The dot product of two vectors a and b (sometimes called inner product, or, since its result is a
   
scalar, the scalar product) is denoted by a ∙ b and is defined as: a  b  a  b  cos 
where ||a|| and ||b|| denote the norm (or length) of a and b, and θ is the measure of the angle between
a and b Geometrically, this means that a and b are drawn with a common start point and then the
length of a is multiplied with the length of that component of b that points in the same direction as
a.
The dot product can also be defined as the sum of the products of the components of each vector:
 
a  b  a x  bx  a y  b y where a and b are vectors of nth dimension; ax, ay are coordinates of a; and
bx, by are coordinates of b.