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. 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.