Physics

Vectors


Determined by an AB-oriented segment, it is the set of all AB-oriented segments.

If we indicate With this set we can symbolically write:

Where XY is any segment of the set.

The vector determined by AB is indicated by or B - A or .
Same vector It is determined by an infinity of oriented segments, called representatives of this vector, which are all equipolent with each other. Thus, a segment determines a set that is the vector, and any of these representatives determines the same vector. Using our capacity for abstraction a little more, if we consider all the infinite oriented segments of common origin, we are characterizing, through representatives, the totality of the vectors of space. Now each of these segments is a representative of a single vector. Consequently, all vectors are represented in that set that we imagine.

The characteristics of a vector they are the same as any of its representatives, that is, the modulus, direction, and direction of the vector are the modulus, direction, and sense of any of its representatives.

The module of is indicated by || .

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)

Sum Sum Properties

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 number by a vector

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

c.v = (ca, cb)

Vector Scalar Product Properties

Whatever are k and c 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

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

There are two unit vectors that form the canonical base for 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:

Note:
To construct 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 be less than v.
If c> 1, then u will be longer than v.
If c <0, then u will have the opposite direction of v.

Vector Decomposition in Single Vectors

To make vector calculations in only one of the planes in which it presents itself, one can decompose this vector into unit vectors in each of the presented planes.

Being symbolized, by convention, î as unit vector of the plane x and as unit vector of the plane y. If the problem to be solved is given in three dimensions, the vector used for the plane z is the unit vector .

So the projection of the vector on the shaft x of the Cartesian plane will be given by , and its projection on the axis y of the plan will be: . This vector can be written as:

=(,), respecting that always the first component in parentheses is the projection in x and the second is the projection on the axis y. If a third component appears, it will be the axis component. z.

In the case where the vector is not at the origin, you can redraw it so that it is at the origin, or discount the part of the plane where the vector is not projected.

Scalar product

Given the vectors u = (a, b) and v = (c, d) we define the scalar product between the vectors u and v as the real number obtained by:

u.v = a.c + b.d

Examples:

The scalar product between u = (3,4) and v = (- 2,5) is:

u.v = 3. (-2) + 4. (5) = -6 + 20 = 14

The scalar product between u = (1,7) and v = (2,3) is:

u.v = 1. (2) + 7. (- 3) = 2-21 = -19

Scalar Product Properties

Whatever the vectors, u v and w and k climb:

Angle between two vectors

The scalar product between vectors u and v can be written as:

u.v = | u | | v | cos (x)

where x is the angle formed between u and v.

Through this last scalar product definition, we can get the angle x between two generic vectors u and v, such as,

as long as none of them are null.