Question

resolve vectors in 3d cartesian coordinates

Answer 1

You may have noticed that we use the same notation to denote a point and to denote a vector. We don't tend to emphasize any distinction between a point and a vector. You can think of a point as being represented by a vector whose tail is fixed at the origin. You'll have to figure out by context whether or not we are thinking of a vector as having its tail fixed at the origin.
Another way to denote vectors is in terms of the standard unit vectors denoted i
i
and j
j
. A unit vector is a vector whose length is one. The vector i
i
is the unit vector in the direction of the positive x
x
-axis. In coordinates, we can write i=(1,0)
i
=
(
1
,
0
)
. Similarly, the vector j
j
is the unit vector in the direction of the positive y
y
-axis: j=(0,1)
j
=
(
0
,
1
)
. We can write any two-dimensional vector in terms of these unit vectors as a=(a1,a2)=a1i+a2j
a
=
(
a
1
,
a
2
)
=
a
1
i
+
a
2
j
.