In details

Vector operations

Sum of Vectors

If v = (a, b) and w = (c, d), we define the sum of v and w by:

v + w = ​​(a + c, b + d)

Vector Sum Properties

I) Commutative: For all vectors u and v of R2:

v + w = ​​w + v

II) Associative: For all vectors u, v and w of R2:

u + (v + w) = (u + v) + w

III) Neutral Element: There is a vector O = (0,0) in R2 such that for every u vector of R2, if have:

O + u = u

IV) Opposite element: For each vector v of R2, there is a vector -v in R2 such that:

v + (-v) = O

Vector Difference

If v = (a, b) and w = (c, d), we define the difference between v and w by:

v - w = (a-c, b-d)

Product of a scalar by a vector

If v = (a, b) is a vector and ç is a real number, we define the multiplication of c by v as:

c.v = (ca, cb)

Vector Scalar Product Properties

Whatever k and ç scalars, v and w vectors:

  • 1 v = v
  • (k c) v = k (c v) = c (k v)
  • k v = c v implies k = c if v is not null
  • k (v + w) = k v + k w
  • (k + c) v = k v + c v

Module of a vector

The modulus or length of the vector v = (a, b) is a nonnegative real number, defined by:

Unit vector

Unit vector is the one with the modulus equal to 1.

There are two unit vectors that form the canonical base into space R2, which are given by:

i = (1.0) j = (0.1)

To build a unit vector u that has the same direction and direction as another vector v, just divide the vector v by its module, that is:


To build a vector u parallel to a vector v, just take u = cv where c is a nonzero scalar. In this case, u and v will be parallel.

If c = 0 then u will be the null vector.
If 0 <c <1 then u will have a length less than v.
If c> 1 then u will be longer than v.
If c <0 then u will have opposite direction to v. Next: Scalar Product