3D Vectors
All of the things we have learned so far can be applied to 3D vectors. Essentially, we add the third dimension so that 3D vectors are now capable of describing points in space rather than just planes. Click here to revise 2D vectors including unit vectors and how to find magnitude and direction. Also recall 2D vector arithmetic and vectors in context such as position vectors and vectors in trigonometry. At this level, we often extend a lot of the earlier mechanics problems to two dimensions – it is very simple to add the third dimension. See Calculus in Kinematics, for example.
Unit 3D Vectors
Recall the unit vectors
and
when working in two dimensions. We now extend to three dimensions and we introduce the third unit vector:
These vectors point in the ,
and
directions respectively. Notice the position of the
and
axes relative to the
-axis. We call this the right-hand rule. This means that if the
-axis is your index finger on your right hand and the
-axis is your middle finger, then the
-axis is your thumb and its points upwards. We often see it in different orientations (see Example 1) but usually with the
-axis pointing upwards. If you find that with your thumb pointing upwards, the
and
axis are on the wrong fingers, you are likely using a left-handed coordinate frame.
With these vectors, we can now identify the position vector that points from the origin to any point in 3D space. The vector
we can see here is a position vector as it points from the origin. We can write
.
It follows, by a simple application of 3D Pythagoras, that the magnitude of this vector is
.
It follows that a unit vector in the direction of is given by
. See Example 1.
3D Vector Arithmetic and Magnitude
We can apply the operations that we saw on the vector arithmetic page in a similar fashion. That is, suppose we have the vectors and
. It follows that
where is a scalar constant
We can easily extend the geometric interpretations that we saw on the vector arithmetic page to 3 dimensions. For example, the vector between two points in 3D is given by
. It follows that the distance between these two points is the magnitude of this vector and given by
. Note that
is not a position vector as it doesn’t point from the origin. It does however lie in the same plane as
and
. We say that
,
and
are coplanar. See Example 1.
We can also extend the concept of direction to 3D vectors. A 3D vector now lies in space and not on a plane but a simple application of SOHCAHTOA (cos=adj/hyp) tells us that
for the vector
that makes an angle of
with the
-axis. Similarly,
and
where
makes an angle of
and
with the
and
axes respectively. See Example 2.