We will see in the next chapter that not only forces but also location vectors  which are vectors pointing from one point in space to another  are fullfledged vectors. And so are velocity and acceleration ( EMch 12 stuff).
In the language of mathematics we address the forces , .... etc. as vectors and use the notation :
If we have numerical values available for the components we would write :
The arrow on top of the symbol merely indicates that this quantity is a vector. Many textbooks use bold/italics symbols instead, like F_{1} because the arrows are costly to print.
On the WWW I am in a similar situation and will use the arrow mode and bold/italics interchangeably.
Using this notation, Equation 3.4c (just above) is then written as
The mathematician would read this as :
Add the vector and to obtain the vector and Equation 3.4c states the rule as to how to do that :
Adding two vectors to each other results in a new vector the xcomponent of which is determined by the adding the xcomponents of the given vectors. (Same for the y and z component)
The physicist and we would say :
The forces and together have the same action on a given body as the force . To obtain one force from the others we perform the same mathematical operations, that is use Equation 3.4c.
How about subtraction of vectors ?
Formally we would write :
but how would we determine the components of if the components of and are given ? Well, a requirement one could put forward here would be the following :
If you subtract first from a vector a vector to obtain an intermediate vector and than ADD to that, you should come back to . The only way how this can be accomplished by using the following rule :
Subtracting two vectors from each other results in a new vector the xcomponent of which is determined by subtracting the xcomponents of the given vectors. (Same for the y and z component) 
If the components of a vector are given, its magnitude (which is always nonnegative) is calculated according to :
If we wish to multiply a given vector, say by a scalar, say "a", we obtain a new vector, say . We formally write :
and the rule which goes along with that is the following :
Multiplying a vector by a scalar results in a new vector the xcomponent of which is determined by multiplying the the xcomponent of the given vector by the scalar. (Same for the y and z component) 
At least for the case of the scalar being a positive integer this rule makes a lot of sense because it ties in nicely with the addition of vectors : whether you add a given vector 4 times to itself or whether you multiply the vector by 4 should give identical results :
Speaking in terms of forces : 4 times is a force which is 4 times as strong as and pointing in the same direction as .
Similarily, in the following equation the left side and right side give identical results :
Speaking in terms of forces : 1 times is a force which has the same magnitude as but is pointing exactly in opposite direction.
Because the magnitude F (and with that 1/F) is never negative the vector
is pointing in the same direction as the vector
itself.
The unique property of
is that its magnitude is always 1 (one)
regardless of what values (and units) the components of
are (except if they are all zero). Because of this
property we call
a unit vector.
Among the infinitely many unit vectors which one can calculate there are a few worth mentioning. If in Equation 4.5a the vector has a positive xcomponent but zero y and zcomponent, that is if it is pointing along the positive xaxis, then is pointing along the positive xaxis as well, still having the length one. In this special case we often (and this is almost universal) give that unit vector the symbol . The analogs for the y and zdirections are given the symbols and , respectively.
The unit vectors , , and can be used to present any arbitrary vector with components F_{x}, F_{y}, and F_{z} in an alternative form :
This way of writing out a vector will come in handy later on in the course, for now it is one example of how addition of vectors and multiplication with a scalar appear in a single equation.
in which "F" and "r" denote the magnitude of and , respectively. is the angle enclosed by the two vectors as shown in Figure 4.6a.

If you know the two magnitudes and the angle you can use Equation 4.6a to determine the value (which may come out positive, negative, or zero) of the dotproduct. 
If you know the components of the two vectors (let's say F_{x}, F_{y},F_{z}, and r_{x},r_{y},r_{z}) there is an alternative way to determine the value of the dotproduct :
Equations 4.6a and 4.6b can be used nicely to calculate unknown
quantities. For example if the components of the two vectors are
given and one wishes to know the angle between them, see
Prob. 4.9e.
In the 2dimensional case it is not too difficult to derive Equation 4.6b
from 4.6a using only elementary geometry, see
Prob. 4.9j.
The vectors and can represent any, even different physical vector quantities.
From the point of view of physics often represents a force acting on a body while represents the distance the same body moves. In this case the value of the dotproduct equals the amount of work the force performs on the body.
Here are two equations which come in handy at times :
Another wellknown case is when the two vectors and are one and the same. We find that for any vector the dotproduct with itself is equal to its magnitude squared :

Lets assume you have given the components of a force
acting on a body of arbitrary shape
at some point B as shown in the figure to the left. The body is supported at point A by what we call a ball/socket joint which allows the body to swivel freely about point A. Lets call the vector from point A to point B the vector and lets assume we know the value of its components. 
Question : About which axis will the given body start to rotate
under influence of the force
and how effective is the given force in trying to actually rotate the
body ?
In Chapter 5 we will do some investigating ourselves, for now I state :
Well, if I gave you the components of these two vectors you could calculate the magnitudes F and r and use the dotproduct to determine the angle between the two vectors and with that the value of M (= moment or torque in physics terms) is determined. The orientation of the axis AC is harder to determine with the tools we have sofar at our disposal.
This is where the crossproduct comes in.
The crossproduct of two vectors is a new vector which is perpendicular to the first two vectors and points in such a direction that it gives you also the direction of rotation. The length of this new vector is given by Equation 4.7a. 
Formally I write :
but the problem is of course to determine the components of the vector for given and .
If you know what the determinant of a matrix is then the following says it all :
or multiplying it all out :
Examples of evaluating the crossproduct of given vectors : Prob. 4.9f , and Prob. 4.9g , and Prob. 4.9h .
Assume for the moment that you have three vectors given, like , , and which from the point of view of mathematics can represent anything (not just forces). The triple product between these three vectors is defined as :
The parenthesis indicate that you first evaluate the crossproduct, which results in a new vector, and then evaluate the dotproduct which results in a scalar "d".
An alternate way to determine the value of "d" is to evaluate the following determinant :
A property of determinants is that exchanging any two rows will merely change the sign (+/) of the answer. As a result you can change the order of the vectors in all together 6 different ways of which I am just showing three :
Click here to do the test.
Prob. 4.10a :
Addition/subtraction of vectors
Prob. 4.10b : Multiplication by a scalar
Prob. 4.10c : Magnitude of Unit vector
Prob. 4.10d : Value of dot product
Prob. 4.10e :
Dot product  Angle between vectors
Prob. 4.10f : Evaluate cross product
Prob. 4.10g : Proof on cross product
Prob. 4.10h :
Cross product  Angle between vectors
Prob. 4.10i :
Broadcast pole, 3D (Prob 3.6h revisited)
Prob. 4.10j :
Proof on dotproduct
Zig Herzog, hgn@psu.edu
Last revised: 09/29/10