Basic Concepts of Vector Algebra

If two vectors are represented in magnitude and direction by two sides of a triangle taken in order, then their sum is \[\overline{\mathrm{AC}}=\overline{\mathrm{AB}}+\overline{\mathrm{BC}}\]

A physical quantity that is described with magnitude alone is called a scalar.

Let ‘l’ be any straight line in plane or three dimensional space. This line can be given two directions by means of arrowheads. A line with one of these directions prescribed is called a directed line in following  first and second fig .

Position Vector:
The three dimensional right handed rectangular coordinate system in  following first fig.

Consider a point P in space, having coordinates (x, y, z) with respect to the origin O(0, 0, 0). Then, the vector  having O and P as its initial and terminal points, respectively, is called the position vector of the point P with respect to O. 
Using distance formula `|vec (OP)| = sqrt(x^2 + y^2 + z^2)`
In practice, the position vectors of points A, B, C, etc., with respect to the origin O are denoted by `vec a , vec b , vec c` etc., respectively in above second fig.

Direction Cosines:
The position vector `vec (OP) (or vec r)` of a point P(x, y, z) as in  following Fig.

The angles α, β, γ made by the vector `vec r` with the positive directions of x, y and z-axes respectively, are called its direction angles. 
The cosine values of these angles, i.e., cosα, cosβ and cosγ are called direction cosines of the vector `vec r` , and usually denoted by l, m and n, respectively.

one may note that the triangle OAP is right angled, and in it, we have `cos alpha = x/r`( r stands for |`vec r`|).  Similarly, from the right angled triangles OBP and OCP, we may write  `cos beta= y/r` and `cos gamma = z/r`.  Thus, the coordinates of the point P may also be expressed as (lr, mr,nr).  The numbers lr, mr and nr, proportional to the direction cosines are called as direction ratios of vector `vec r` , and denoted as a, b and c, respectively.

A physical quantity that is described with both magnitude and direction is called a vector.

If two co-initial vectors are represented in magnitude and direction by adjacent sides of a parallelogram, then their sum is represented in magnitude and direction by the diagonal of the parallelogram passing through the common point.

∴ \[\overline{\mathrm{AC}}=\overline{\mathrm{AB}}+\overline{\mathrm{BC}}\]

A vector whose magnitude is zero is called a zero vector.

A vector that has the same magnitude as a given vector but acts in the opposite direction is called a negative vector.

Two vectors having the same magnitude and the same direction are called equal vectors.

A vector that describes the position of a point with respect to the origin is called a position vector.

A vector that has a magnitude of one unit and is used to indicate direction is called a unit vector.

Vectors that act in the same plane are called coplanar vectors.

Vectors that are perpendicular to each other are called orthogonal vectors.

When a vector \[\vec P\] is split into two mutually perpendicular parts along the horizontal and vertical axes, those parts are called rectangular components.

For any two vectors \[\vec P\] and \[\vec Q\]​:

The commutative law holds true for addition of vectors but not for subtraction.

For three vectors \[\vec P\], \[\vec Q\]​, and \[\vec R\]:

The associative law holds true for addition of vectors but not for subtraction.

Two vectors can be added using either the Triangle Law or the Parallelogram Law. When two vectors \[\vec P\] and \[\vec Q\]​ are represented as two sides of a triangle (or two adjacent sides of a parallelogram), their resultant is represented by the third side (or the diagonal).

The resultant R of two vectors \[\vec P\] and \[\vec Q\]​ always lies between their difference and their sum:

If two vectors \[\vec P\] and \[\vec Q\]​ act at an angle θ, the magnitude of their resultant is:

If a vector \[\vec P\] is resolved into two rectangular components:

• Horizontal component: P_x = P cos⁡ θ
• Vertical component: P_y = P sin ⁡θ

\[\vec P\] ⋅ \[\vec Q\] ​= PQ cos θ

∣\[\vec P\] × \[\vec Q\]​∣ = PQ sin θ