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