Two Dimensional Vectors (Level 5) In this video we will go over various examples
that make use of vector operations, specifically vector addition, vector subtraction and scalar
multiplication. We will solve these problems by using the geometric interpretation of these
particular vector operations. Alright let’s go over the first example.
Write each combination of vectors as a single vector.
Alright here we have line segments connected by various vertices. Since these are line
segments we are free to assign the direction of vectors by appropriately denoting the initial
and terminal points. Let’s start with the first example.
Here we are asked to find the sum of vector CB and vector BA. First let’s identify each of
the vectors on the diagram, vector CB is located here and vector BA is here. Next we go ahead
and add the vectors geometrically by applying the triangle law. Notice that the tail of
vector BA is aligned with the head of vector CB. All that is left to do is to draw and
determine the resultant vector; in this case the vector will extend from the tail of vector
CB towards the head of vector BA. Looking at the diagram we see that this vector can
be denoted as vector CA and this is the resultant vector.
Alright let’s try the next example, here we are asked to add vector AC and vector CD,
like in the previous example let’s go ahead and apply the triangle law for vectors, notice
that the tail of vector CD is aligned with the head of vector AC this means that the
resulting vector is going to have an initial point at A and a terminal point located at
D, in other words the resultant vector would be vector AD.
Let’s try the next problem, here we are asked to find the difference between vector
BD and vector CD. Let’s first identify vector BD and vector CD. Now in order to carry out
this difference we can think about this problem as a special case of adding vectors in this
case we are adding the negative of vector CD to vector BD. So if we were to take vector
CD and align its tail to the head of vector BD we would obtain the following, now we need
to flip vector CD so that it points in the opposite direction as follows. Now we go ahead
and apply the triangle law, notice that the resultant vector in this example would be
equal to vector BC.
Alright let’s try the next example, here we are asked to find the sum of vector AD,
vector DC and vector CB. When adding more than 2 vectors the procedure is essentially
the same, you want to add the vectors by using the tail-to-head method, you can think of
this process as applying the triangle law multiple times. In the end the resultant vector
will have its initial point at A and terminal point at B, thus the resultant vector would
be vector AB. Alright let’s move along and try a different
type of problem. Given vectors u, v and w draw and label the
resultant vector of the following vector expressions. Ok in this problem we are provided with 3
distinct vectors each with their own magnitude and direction. The first vector expression
is asking us to find and draw the resultant vector when vector u and vector v are added.
We can geometrically find the sum by applying the triangle law, so we go ahead and align
the tail of vector v with the head of vector u as follows; the resultant vector will then
have its initial point located at the tail of vector u and its terminal point at the
head of vector v. Alternatively, we can also make use of the parallelogram law and align
the tails of both vectors together and then make copies of each vector that are parallel
to one another forming the sides of the parallelogram. Using this method we can draw the resultant
vector by starting at the intersection where the tails of both vectors coincide and extending
the directed line segment along the diagonal of the parallelogram. Regardless of what method
you apply you should obtain the same resultant vector.
Ok let’s try the next example, here we are asked to find the resultant vector when we
add vector v and vector w. Keep in mind that it really does not matter which vector you
add first as long as you follow the triangle law or parallelogram law everything should
workout. Using the triangle law we would obtain the following resultant vector. In the same
manner if we use the parallelogram law we would obtain the same exact resultant vector.
Alright let’s try the next example; here we need to find the resultant vector when
we subtract vector v from vector u. The procedure for subtracting vectors is essentially the
same as adding vectors; all we need to do is to make sure we flip the direction of the
vector being subtracted. In this case we first align the vectors by making use of the head-
-to-tail method or triangle law, then we take vector v and flip it in the opposite direction,
from here the final step is to draw the resultant vector, this vector is drawn just like in
the previous examples. We can also use the parallelogram law, when using this method
always make sure you flip the negative vector before applying the parallelogram law.
Alright let’s move along to the next example, here we are asked to add all three vectors,
the best way to tackle these types of problems is by applying the triangle law multiple times,
so we essentially need to align the head of one vector with the tail of another vector
one after the other as follows, the resultant vector will have its initial point located
in the tail of vector u and will end at the head of vector w, as follows.
Let’s try the next example, here we are asked to carry out a scalar multiplication
of vector w, recall that scalar multiplication scales a vector by changing its length or
in the case of negative scalars it can also change the direction of a vector. In this
example we are asked to draw vector 2 times w, we can draw this expression by taking vector
w and doubling the size or magnitude of the vector as follows, keeping the same direction
as the original vector. Ok let’s try the next problem, here we are
once again asked to carry out a scalar multiplication, in this example we are asked to scale vector
u by half its size, we also have a negative sign so we also need to change the vectors
direction so that it points in the opposite direction as follows.
Alright let’s try the next example, here we are asked to subtract 2 vectors in this
case we are asked to find the difference between vector u and 3 times vector v. Similar to
vector addition we need to align the vectors by using the triangle law, in this particular
example we first need to scale vector v by a factor of 3, then we go ahead and align
the tail of vector 3v with the head of vector u and then we flip vector 3v in the opposite
direction as follows. The resultant vector can then be drawn by connecting the initial
point of vector u with the terminal point of vector 3v.
Ok let’s go over the final example, here we are asked to carry out a scalar multiplication,
a sum and difference of vectors. Let’s draw the scalar multiplication of each vector,
for vector u we double the length, for vector v we scale the vector by a third of its original
length and for vector w we triple its original length. Next let’s go ahead and add vector
2u with vector one third v by using the tail to tip method, next let’s go ahead and take
vector 3w and flip it in the opposite direction then we go ahead and align the tail of this
vector with the tip of vector one third v. The resultant vector would then be drawn from
the tail of vector 2u to the tip of vector 3w.
Alright, now that we have a better understanding of the geometry of scalar multiplication,
vector addition and vector subtraction. We are ready to learn how to carry out vector
operations algebraically. In our next video we will learn a more powerful and precise
method for computing various vector operations.