For two arbitrary vectors A and B, expressed as Cartesian vectors, their dot product is a scalar and is defined algebraically as the sum of the 
products of their respective i, j and k 
components.
Geometrically, even if the two vectors are not on the same plane, they can be parallel-transported to be concurrent,
and they form an angle of theta.
Their dot product equals to the magnitude of the 
projection of vector A on B, A prime, multiplied 
by the magnitude of vector B,
or the magnitude of the projection of vector B on 
A, B prime, multiplied by the magnitude of A. 
Either way the dot product equals to
magnitude A, times magnitude B, times cosine 
theta.
Because of the algebraic and geometric 
definitions of dot product, 
dot product can now be used to find the angle 
theta between the two vectors A and B following 
this formula.
Or we can use dot product to find the projection 
vector of any vector along a specified axis.
Note, u_A is the unit vector along the a-a axis.
Let's look at this example. Two forces are drawn in the xyz coordinate system, 
and they are also expressed in their respective Cartesian vector forms,
given in the unit of Kip, kilo-pound,
we need to determine the angle between them as well as the magnitude of the projection force of F_1 along the line of action of F_2.
For this problem we are going to use the dot product. But first let's calculate their magnitudes.
Now for the dot product we're going to use the algebraic equation
by finding the sum of the products of their respective coordinates,
to be negative 32.6. Notice that it does have a unit of kip squared.
Since now we know the dot product as well as the magnitudes,
for angle theta, it can be calculated as
inverse cosine or arccosine of the dot product of these two force vectors, divided by their respective magnitudes,
to be 118°.
and then for the projection force F_1 along the line of action of F_2, it simply equals to 
the magnitude of F_1 multiplied by cosine theta. And the absolute value is 3.4 kip.
And this answers this question, but let's also look at an alternative way to solve this problem.
With the magnitudes of F_1 and F_2,  alternatively, we can calculate the unit vectors of these two force vectors.
And since the angle made by these two unit vectors is the same as the angle made by the original two force vectors, therefore angle theta 
can be easily determined as inverse cosine of the dot product of these two unit vectors.
And you'll get the same answer.
With the unit vector of F_2 determined,
the projection force of F_1 along the line of action of F_2 simply equals to the dot product of F_1
and the unit vector u_F2.
You will get the same answer,
however this method is convenient when you do not need to determine the angle theta.
