Three Dimensional Vectors Level 1 In the previous videos we reviewed the basics
of two dimensional vectors also known as planar vectors. In the following series of videos
we go over three dimensional vectors, also known as vectors in space. The transition
from planar vectors to vectors in space requires the addition of a third component in order
to represent the direction that points along the z-axis of a three dimensional coordinate
system. For example the position vector with an x-component
of 2 and a y-component of 3 is represented in a two dimensional coordinate system as
follows. Now if we wanted to represent this same vector in 3D space we would need to add
an additional component in this case the z-component. For this particular example the vector is
going to be located on the xy-plane so it will have a z-component of 0. So this vector
will have an x-component of 2 , and a y-component of 3 and a z-component of 0. In the same manner
the following vectors would be located on the xz-plane and yz-plane respectively, notice
that in each case there is a component equal to 0. Any vector in R cubed that has one component
equal to 0 will lie on a coordinate plane because they resemble two dimensional vectors.
In general a vector located in space will have 3 components and are denoted by ordered
triples as oppose to ordered pairs. If two components of a vector in R cubed are equal
to zero then the vector will lie along a line, for example the following vectors each have
2 components equal to 0, and they lie in the direction of the x-axis, y-axis, and z-axis
respectively. Just like two dimensional vectors can be represented by using the standard unit
vectors (also known as standard basis vectors), we can also use the same notation to denote
vectors in space. In this case we will use the unit vector i hat, j hat, and the unit
vector k hat which represents the unit vector that points along the z-axis.
Any vector in space can be represented by a scalar multiple and sum of the unit vectors.
For example the vector with components 1, 2 and 3 can be represented as i plus 2j plus
3k. If a vector has zero for every single component it is referred to as the zero vector
and can be represented by a single point. Now if a vector is not in standard position
with the tale located at the origin and it is instead represented by two points in space
like point P for the tale or initial point and point Q for the head or terminal point,
then we can find the component form of vector v by subtracting the coordinates of the initial
point from the coordinates of the terminal point as follows.
Lastly, many of the properties for two dimensional vectors also apply to three dimensional vectors
for, example two vectors are still equal if and only if the components of both vectors
are equal in other words vector u is equal to vector v if and only if the x-components,
y-components and z-components are equal. Two nonzero vectors are still parallel if
there is some scalar c such that vector a equals the scalar c times vector b. In other
words the x, y and z-components of vector a are a constant multiple of the x, y and
z-components of vector b respectively. The length or magnitude of a vector in space
is calculated similarly to planar vectors the only difference is that we include the
third component into the expression as follows, this is essentially the Pythagorean theorem
in three dimensions. To find a unit vector in the direction of
a vector v we simply take the components of vector v and divide it by the magnitude of
vector v, as long as vector v is not the zero vector.
Lastly the properties of vector addition and scalar multiplication given for planar vectors
are also valid for vectors in space. For vector addition we simply add the vectors component
wise, and when multiplying a scalar by a vector we simply multiply this scalar to each of
the vectors components. Alright let’s go over some examples and
illustrate how to solve various problems involving three dimensional vectors.
Find the component form of vector v and write the vector using standard unit vector notation.
Sketch vector v with its initial point at the origin.
Alright here we are given the initial and terminal points of a vector, in order to find
the component form of vector v we simply need to subtract the coordinates of the initial
point from the coordinates of the terminal point, doing that we obtain the following
values for the components of vector v. Next we go ahead and write this vector using
the standard unit vectors. This is a pretty straight forward process since each of the
components corresponds to i, j and k respectively. Doing that we obtain the following expression.
Finally we need to sketch this vector in standard position so we start at the origin and use
the components to plot the coordinates of the terminal point just the way you plot points
in a three dimensional coordinate system. So we go ahead and move 4 units along the
positive x-axis and move 5 units along the negative y-axis and move 2 units along the
positive z-axis. Then we go ahead and join these two points with a directed line segment
as follows. Alright let’s go over the next example.
Find the component form, and magnitude of the vector with initial point (4, -5, 2) and
terminal point (-1,7,-3). Then find a unit vector in the direction of this vector.
Alright similar to the previous example we are given the initial and terminal points
of a vector in three dimensions. Let’s call this vector vector v. So we go ahead and subtract
the coordinates of the initial point from the coordinates of the terminal point as follows.
This expression represents the components of vector v. Now let’s find the magnitude
or length of vector v, this requires an application of the Pythagorean theorem in three dimensions,
substituting the components into the expression and simplifying we obtain the square root
of 194 as the magnitude of vector v. Alright now that we found both the component
form and the magnitude of vector v we can go ahead and find a unit vector in the direction
of vector v. We find this unit vector by simply dividing each of the components of vector
v with its magnitude doing that we obtain the following components for the unit vector
in the direction of vector v. You can easily verify that it’s a unit vector by computing
the magnitude and checking that it is equal to 1. Alright let’s go over the final example.
Vector v has components equal to <1, -2/3 and 1/2>. The initial point of vector v is (0,
2, and 5/2). Find the terminal point. Here we are given the components of vector
v and its initial point. We are asked to find its terminal point. We can figure out the
terminal point by using the expression for finding the components of a vector and working
backwards. We know that if we were to take the initial
point and subtract each coordinate with the coordinates of the terminal point we should
obtain the component form. With this set up we can see that finding the terminal point
essentially requires us to solve for the x, y and z-coordinate of the terminal point.
Isolating each expression and solving for the coordinate point we obtain 1, 4/3 and
3 as the coordinates of the terminal point. Alright in our next video we will go over
slightly more challenging examples.