See the triangle rule and the parallelogram rule.
There is an associative law of vector addition:
(u + v) + w = u + (v + w)
and an associative law of scalar multiplication:
a(bu) = (ab)u.
Here u, v and w are any vectors and a and b are any scalars.
The Cartesian form of a vector u in three dimensions is a representation of u as a sum of vectors in the directions of the standard x, y and z axes:
u = ai + bj + ck
where i, j and k are unit vectors in the positive x, y and z directions, respectively, and a, b and c are the Cartesian components of u.
In two dimensions (in the xy plane), the Cartesian form of a vector v is
v = ai + bj.
This is a law of vector addition: u + v = v + u, for all vectors u and v.
These are laws connecting vector addition with multiplication by scalars:
(a+b)u = au + bu.
and
a(u + v) = au + av.
Two (free) vectors are equal if they have the same direction and magnitude (length). They do not need to have the same point of application.
This rule expresses a vector 
as the difference
of the position vectors for the head B and the tail A.
The magnitude of a vector is its length.
The moment of a line vector (l,v) about a point O is the vector product of the position vector of any point P on l with v.
There three ways to multiply vectors:
The negative of a vector is the vector with the same length but opposite direction as the given vector.
The negative of the zero vector is the zero vector.
Two vectors u and v are said to be orthogonal if their directions are at right angles; that is, the scalar product u · v is 0.
This is a rule for vector addition. The sum of two vectors is the diagonal of the parallelogram that has the two given vectors as its sides.
Two vectors are said to be parallel if they have the same or opposite direction or if one of them is the zero vector.
Given points A and B, the position vector
is the vector with tail at
A and head at B; that is, it starts at A and goes to B.
A scalar is a quantity, such as time, specified by a single number.
The scalar product is 0 if and only if the vectors are perpendicular.
Subtraction can be defined as the addition of the negative of a vector. That is, if u and v are vectors, the result of subtracting v from u is u + (-v).
This is a rule for vector addition. The sum of two vectors is the third side of the triangle that has the two given vectors as its sides.
A vector whose magnitude (i.e., length) is equal to 1 is called a unit vector. There are exactly two unit vectors in any given direction and one is the negative of the other.
A vector is a quantity, such as force, with a magnitude (i.e., length) and a direction.
The magnitude of a vector is a scalar quantity.
The vector product of vectors u and v is the vector whose length is the product of the lengths of u and v multiplied by the sine of the angle between them.
The direction of the vector product is perpendicular to u and v and points in the direction of your thumb if your right hand curls from u to v.
The vector product is 0 if and only if the vectors are parallel.
The vector whose magnitude (that is, length) is equal to 0 is called the zero vector. There is no direction associated with the zero vector. This is the only exception to the statement that a vector has both magnitude and direction.