Operations with vectors
Vector addition and subtraction
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Given the vectors a a x , a y  and b bx , b y  the sum of a and b is
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a  b  a x  bx , a y  b y 
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Given the vectors a a x , a y  and b bx , b y  the difference of a and b is
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a  b  a x  bx , a y  b y 
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.
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
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Given the vector a a x , a y  and the real number  the product of
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a  a x , a y 
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a by  is
The length of a is |||a|. If the scalar is negative, it also changes the direction of the
vector by 180o. Two examples ( = -1 and  = 2) are given in the
picture.
Here it is important to check that the scalar multiplication is
compatible with vector addition in the following sense: (a + b) =
a + b for all vectors a and b and all scalars .
One can also show that a - b = a + (-1)b.
k=1
k>1
Note: Multiplication of a non-zero free vector by a real number has
the following effects:
k < –1
stretch and reverse direction
k = –1
reverse direction
–1< k <0
shrink and reverse direction
k=0
becomes zero vector
0< k <1
shrink
do nothing
stretch
Note: In mathematics, numbers are often called scalars to distinguish them from
vectors.
Dot product
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Given the vectors a a x , a y  and b bx , b y  the dot product (sometimes called inner
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product or scalar product) of a and b is a  b  a  b  cos  . It can be also defined
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as a  b  a x  bx  a y  b y
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where a and b denote the norm (or length) of a and b and θ is the measure of the
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angle between a and b .
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Geometrically, this means that a and b are drawn with a common start point and then
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the length of a is multiplied with the length of that component of b that points in the
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same direction as a .
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.
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To normalize a vector a a x , a y  , scale the vector by the reciprocal of its
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a
length a . That is: u a  
a