Vectors in a Plane and  Space 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 in the coordinate system Angle between vectors in a coordinate plane Scalar product of vectors examples
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.
Scalar product in the coordinate system
The scalar product can be expressed in terms of the components of the vectors,  - the scalar product in three-dimensional coordinate system, - the scalar product in the coordinate plane.
Angle between vectors in a coordinate plane
From the scalar product  a · b = | a | · | b | · cosj  derived are, the angle between two vectors, formulas
 - for plane vectors - for space vectors  The cosine of the angle between two vectors,  a and b represents the scalar product of their unit vectors, Scalar product of vectors examples
Example:   Applying the scalar product, prove Thales’ theorem which states that an angle inscribed in a
semicircle is a right angle.
 Solution:  According to diagram in the right figure  since square of a vector equal to square of its length,  thus,  a and b are orthogonal vectors as .
Example:   Prove the law of cosines used in the trigonometry of oblique triangles.
 Solution:  Assuming the directions of vectors as in the right diagram Using the scalar product and substituting square of vectors by square of their lengths, obtained is a2 = b2 + c2 - 2bc · cosa  - the law of cosines. In the same way, Example:   Determine a parameter l so the given vectors, a = -2i + l j - 4k  and  b = i - 6 j + 3k  to be perpendicular.
Solution:  Two vectors are perpendicular if their scalar product is zero, therefore Example:   Find the scalar product of vectors, a = -3mn and  b = 2m - 4n if  | m | = 3 and  | n | = 5 , and the angle between vectors, m and n is 60°.
 Solution: Example:   Given are vertices, A(-2, 0, 5), B(-3, -3, 2), C(1, -2, 0) and D(2, 1, 3), of a parallelogram,
find the angle subtended by its diagonals as is shown in the diagram below.
 Solution:    Example:   Given are points, A(-2, -3, 1), B(3, -1, -4), C(0, 2, -1) and D(-3, 0, 2), determine the scalar and vector components of the vector AC onto vector BD.
Solution:  The scalar component of the vector AC onto vector BD, The vector component of the vector AC in the direction of the vector BD equals the product of the scalar component ACBD and the unit vector BD° that is    Vectors in 2D and 3D Contents 