The three dimensional rectangular coordinate 
system, also known as the Cartesian coordinate 
system,
is made of three number lines
that are concurrent, meaning that they intercept 
each other at the origin,
and also perpendicular to each other.
These three number lines, known as the x, y 
and z axes, divide the space into eight octants,
as demonstrated by the eight cubes with 
different colors in this image.
The reason why this is called a right-handed coordinate system is because the positive 
directions of the three axes follow the right hand 
rule,
which means that if you roll the four fingers in 
your right hand from the positive x direction 
towards the positive y direction,
as shown in this image, then your thumb will 
point towards the positive z direction.
So if we transfer an arbitrary vector A into an 
established Cartesian coordinate system,
with the tail of the vector fall on the origin,
then we can first apply the parallelogram law to 
resolve vector A into two component vectors A_z 
that falls along the z axis, and A prime that 
falls within the x-y plane. Then we can apply the 
parallelogram law again
to resolve vector A prime into the two 
components,
A_x along the x axis and A_y along the y axis.
We can prove that A equals to A_x plus A_y 
plus A_z, and therefore A_x, A_y and A_z are 
the three rectangular components of vector A.
Since a vector needs to be described by two 
parts, its magnitude and its direction,  
we can separate these two parts by defining unit 
vectors.
For any arbitrary vector A, its unit vector u_A 
has the same direction but a magnitude of unit 
length one.
Therefore vector A can be expressed by its 
magnitude A, 
which is a scalar, multiplied by its unit vector 
u_A.
As mentioned earlier, a unit vector can be 
defined for any arbitrary vector A,
however there are three special unit vectors, i, j 
and k, which as you can tell have their own 
names.
They are special because they are designated 
to the directions of x, y and z axes respectively 
in the Cartesian coordinate system.
Therefore, using the unit of vectors i, j and k, the 
component vectors along the x, y and z axes 
can now be written as 
A_x i,  A_y j and A_z k, with A_x, A_y and A_z 
being the magnitudes of the component vectors.
When the vector is expressed in this way we 
call it a Cartesian vector.
And by applying the Pythagorean theorem twice 
we can derive that the magnitude of the vector A 
equals to
the square root of A_x squared plus A_y 
squared plus A_z squared.
To describe the direction of the vector we can use the coordinate direction angles: alpha, beta and gamma.
Alpha is defined as the angle between the vector 
and x axis, and similarly beta and gamma are
angles between the vector and the y and z axes 
respectively.
If we highlight the plane made by the vector A and the x axis, shown as the blue shaded area, then according to trigonometry 
we know that cosine alpha equals to A_x divided by A.
Similarly, as shown in the right triangle made by 
the vector and the y axis, cosine beta also 
equals to A_y divided by A.
And again, cosine gamma also equals to A_z 
divided by A.
And then because the unit vector of vector A, u_A, equals to vector A divided by its magnitude,
u_A can be derived as A_x over A times i, plus 
A_y over A times j, plus A_z over A times k. Or 
u_A equals to 
cosine alpha i, plus cosine beta j, and plus 
cosine gamma k.
And since the magnitude of unit vector is always 
one, therefore we can come to the conclusion 
that the sum of the cosines squared of
the three coordinate direction angles for any 
Cartesian vector must equal to one.
Therefore when you perform Cartesian vector 
addition or subtraction, all you need to do is to 
sum the i, j and k components of the vectors up 
separately,
and those will become the new i, j and k 
components of the resultant vector.
Let's look at this example. We are given a force vector in Cartesian vector form, and we need to determine its magnitude, its unit vector, 
as well as the three coordinate direction angles, alpha, beta and gamma.
Let's look at the magnitude first.
The magnitude of any Cartesian vector simply equals to the square root of the sum of the squares of each component.
Therefore from here we can easily calculate the magnitude to be 4.2 kilonewton.
And then to determine the unit vector:
the unit vector of any vector equals to the vector divided by its magnitude,
and as you can see the unit kilonewton gets cancelled out, and the unit vector u_F
is a dimensionless quantity. You can simplify and if you want to.
And then lastly, for the direction angles,
remember the x, y and z components of the unit vector are the cosine values of the three angles alpha, beta and gamma, therefore angle alpha,
which is the angle made by the vector with the positive direction of the x axis, equals to the inverse cosine or arccosine 2/7,
to be 73.4°. And similarly beta equals to arccosine the y component of the 
unit vector -3/7, and also gamma equals to arccosine 6/7.
And now we calculated the magnitude, unit vector and the direction angles, and that answers this question.
In this example a force with the magnitude of 1200 newton is shown in the xyz coordinate system, and we are also given
two of its direction angles, angle alpha is 60° and beta is 45°.
We need to express this force in Cartesian vector form and also determine its unit vector.
Since we know that for any Cartesian vector its three direction angles alpha, beta and gamma 
are related by this equation, therefore we can try to solve for gamma.
Cosine gamma squared equals to 0.25,
and cosine gamma has two possible values, positive or negative 0.5, which means that gamma can be either 60° or 120°.
Now let's look at the image. As you can see the force falls in the octant made by the positive x, positive y and negative part of the z axis.
And also we know that the direction angle is defined as the angle made by the force with the positive part of the axis.
Therefore this angle gamma is 120°.
From here we can determine its unit vector by plugging in the values of cosine alpha, beta and gamma,
and then we can determine the Cartesian form of this force vector by multiplying its magnitude
to its unit vector.
As you can see the unit vector is dimensionless but the force vector does have the unit of newton. And this answers the problem.
