Two Dimensional Vectors Level 9 In this video we will cover unit vectors,
standard unit vectors and direction of vectors. Let’s first talk about unit vectors.
If the magnitude of a vector happens to be equal to 1 we call the vector a unit vector
these vectors are going to be extremely useful as we cover more complicated concepts associated
with multivariable calculus. In many applications involving vectors, it is very useful to find
a unit vector that has the same direction as a given vector. This way we can extract
the direction of a vector and use it in various applications.
If vector v is a nonzero vector in a plane, then vector u can be found by taking vector v and
dividing it by the magnitude of vector v, this new vector u will have a length equal
to 1 and will have the same direction as vector v, as long as vector v does not equal zero. We can denote
the unit vector by using a bold lower case u or we can draw a “hat” symbol directly
above letter u. This symbol let’s you know that this particular vector represents
a unit vector. Let’s proof that the unit vector u has a
length or magnitude of 1 and that it points in the same direction as vector v. Let’s
start with proving that the unit vector u has the same direction as the original vector v.
We can show that the unit vector u has the same direction as vector v by noticing that
one over the magnitude of vector v is essentially a positive scalar so it will not reverse the
direction of the original vector v, in addition we are taking this scalar and multiplying
it by vector v. Recall that a scalar multiple of a vector v, generates vectors that are
parallel to vector v which means that vector u has the same direction as vector v.
Next let’s show that the length or magnitude of vector u is equal to one, we can show this
by finding the magnitude of the unit vector u. We first replace vector u with its equivalent
expression in this case one over the magnitude of vector v times vector v. This is essentially
the magnitude of a scalar times a vector so we make use of the length of a scalar multiple
property from the previous video and rewrite the expression as follows, the constant on
the left stays positive so we can rewrite it as follows, and finally notice that we
have the magnitude of vector v over the magnitude of vector v which cancel out, giving us 1
as our final answer. In essence all you need to do in order to
find a unit vector is to find the magnitude of the vector and divide it to each of the
components of the vector. This will generate a vector with length equal to 1, while maintaining
its direction. This will be very useful, later on in this course. The final things to know
about unit vectors is that the unit vector u is usually called a unit vector in the direction
of v, and the process of multiplying a vector v by 1 over the magnitude of v to get a unit
vector is called the normalization of vector v. Alright let’s go over a numerical example
Find a unit vector in the direction of v and
verify that it has a length of 1. Let’s go ahead and find a unit vector
in the direction of vector v, all we need to do is to find the magnitude of vector v
and divide it into each of the components of vector v. The magnitude of vector v is
equal to the square root of 29, now it is just a matter of distributing the scalar 1
over the square root of 29 to each of the components as follows. The unit vector will
now have an x-component of negative 2 over the square root of 29 and y-component of 5
over the square root of 29. In order to verify that this vector is indeed
a unit vector all we need to do is to find the magnitude, so we go ahead and square each
of the components, add them and then take the square root of the result. Doing this
we see that we obtain 1 as our magnitude, this shows that the unit vector u does indeed
have a magnitude or length equal to 1. Now that we are familiar with unit vectors,
it is time to talk about a special type of unit vectors. Specifically the unit vectors
with components one zero and zero one, these unit vectors are called the standard unit
vectors in a plane and are denoted as follows. On paper we usually denote these unit vectors
with a lower case letter and draw the “hat” symbol on top. This is pronounce as “i hat”
and “j hat” respectively. One vector points in the direction of the x-axis in this case
i hat and one vector points in the direction of the y-axis in this case j hat. Both vectors
have a length or magnitude of 1. Standard unit vectors are used to represent
any vector uniquely, for example we can represent vector v in terms of these standard unit vectors,
notice that the x-component of vector v can be constructed by adding 3 i hat vectors,
in the same manner the y-component of vector v can be constructed by using 5 j hat vectors.
Mathematically we can denote vector v by using standard unit vectors as follows. Vector v
can be represented as the sum of the x-component plus the sum of the y-component, and each
component can be represented as a scalar multiple of the standard unit vectors, we simplify
this expression by replacing the unit vectors with i hat and j hat respectively. This means
that any vector in a plane can be expressed in terms of standard unit vectors. In this
example the expression 3 i hat and 5 j hat is a called a linear combination of i and
j where the scalars 3, and 5 are the horizontal and vertical components of v respectively.
In general any vector in a plane can be expressed by a linear combination of the standard unit
vector i and j. You can think of standard unit vectors as another way of denoting the
components of a vector. Alright now let’s take a look at another way of describing vectors
in a plane. Another way of expressing vectors is by describing
the direction of a vector by using angles. Recall from trigonometry that a unit circle
was used to define trigonometric functions. The unit circle has a radius equal to one,
we also defined an angle theta, in standard position as an angle that started in the positive
x-axis and rotated counter clockwise intersecting a point (x, y) of the unit circle. In addition,
we are able to construct right triangles where the lengths of the legs represented the x-coordinate
and y-coordinate of this point. By using right angle trigonometry we are able to express
the x-coordinate in terms of the angle theta as cosine of theta and the y-coordinate as
sine of theta. Now let’s take a look at the radius of the
unit circle and convert it into a vector, this vector will be a unit vector since the
length or magnitude of this radius is equal to 1. We will denote this unit vector as u
hat. If u hat is a unit vector and theta is the angle measured counterclockwise from the
positive x-axis to u hat, then the terminal point of u hat lies on the unit circle, the
components of this unit vector will equal to, cosine of theta i hat plus sin of theta
j hat. This is another way of denoting a unit vector whose direction is expressed in terms
of an angle. We can extend this definition to any nonzero vector v that makes an angle
theta with the positive x-axis, the direction in this case will be the same as the unit
vector the only difference is that the vector will be scaled by the magnitude of vector
v, we can denote vector v as follows, vector v equals the magnitude of vector v times cosine
of theta I hat, plus the magnitude of v times sin of theta j hat.
For example say I have vector v with a magnitude of 3 and it also makes an angle of 30 degrees
or pi over 6 radians with the positive x-axis. To write vector v as a linear combination of the
standard unit vectors i hat and j hat. We simply take the magnitude of vector v in this
case 3 and multiply it by cosine of pi over 6 to obtain the horizontal or “i hat”
direction, and multiply 3 by sin of pi/6 to obtain the vertical or ‘j hat’ direction.
Simplifying the expression we obtain the following: 3 times the square root of 3 over 2 “i hat”
plus 3 over 2 “j hat”. When evaluating trigonometric functions make sure your calculator
is in the correct mode either degrees or radians. Alright in our next video we will go over
various examples that make use of unit vectors.