Vectors in a Plane and  Space
Vectors and a coordinate system, Cartesian vectors
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
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.
Vectors examples
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
Pre-calculus contents I