Vectors in a Plane and  Space Vectors in three-dimensional space in terms of Cartesian coordinates Scalar product or dot product or inner product Orthogonality or perpendicularity of two vectors
Different positions of two vectors and the corresponding values of the scalar product Square of magnitude of a vector Scalar product of unit vectors Scalar or dot product properties
Scalar product or dot product or inner product
On the beginning of this section we have already mentioned that there are physical quantities as; force, velocity, acceleration, electric and magnetic field and so on, which all have vectors’ properties.
There’s another group of physical quantities as; distance, time, speed, energy, work, mass and so on, which
have magnitude but no direction, called scalar quantities or scalars.
The word scalar derives from the English word "scale" for a system of ordered marks at fixed intervals used in measurement, which in turn is derived from Latin scalae - stairs.
The magnitude of any vector is a scalar.
The scalar (numeric) product of two vectors geometrically is the product of the length of the first vector with projection of the second vector onto the first, and vice versa, that is   The scalar or the dot product of two vectors returns as the result scalar quantity as all three factors on the right side of the formula are scalars (real numbers).
The result will be positive or negative depending on whether is the angle j between the two vectors are acute or obtuse.
For example, in physics mechanical work W is the dot product of force F and displacement s, Obviously, a change of the angle between the two vectors changes the value of the work W, from the maximal value for j = 0°   =>    W = | F | · | s |, to the minimal value for j = 180°   =>    W = - | F | · | s |. For j = 90° the force F does any work on an object, since It is only the component of the force along the direction of motion of the object which does any work.
Orthogonality or perpendicularity of two vectors
Therefore, if the scalar product of two vectors, a and b is zero, i.e., a · b = 0 then the two vectors are orthogonal.
And inversely, if two vectors are perpendicular, their scalar product is zero. Different positions of two vectors and the corresponding values of the scalar product
Different positions of two vectors and the corresponding values of their scalar product are shown in the below figures.       Square of magnitude of a vector
The scalar product of a vector with itself,  a · a = a2  is the square of magnitude of a vector, that is thus, - in the coordinate plane, and - in 3D space.
Scalar product of unit vectors
The unit vectors, i, j and k, along the Cartesian coordinate axes are orthogonal and their scalar products are,  Scalar or dot product properties
 a)  k · (a · b) = (k · a) · b = a · (k · b),  k Î R b)  a · b = b · a c)  a · (b + c) = a · b + a · c d)  a · a = a2 = | a |2 According to the definition of the dot product, from the above diagram, then what confirms the distribution law.

The associative law does not hold for the dot product of more vectors, for example

a · (b · c) is not equal (a · b) · c
since  a · (b · c) is the vector a multiplied by the scalar  b · c,  while (a · b) · c  is the vector c multiplied by the scalar a · b.    