Two Dimensional Vectors (Level 4) In this video we will learn about vector operations
this is also known as the vector arithmetic. Many problems that make use of vectors require
slightly more complicated calculations other than just finding the magnitude of a vector.
In order to carry out these calculations we need to learn algebraic operations with vectors
these vectors operations are carried out similar to algebraic operations with real numbers.
Before we start we need to remember that a vector represents a directed line segment
with a specific length known as the magnitude and a specific direction. We will be using
this geometric representation of a vector to illustrate various vector operations geometrically.
You can think about this vector as representing the displacement of a particle that starts
at one point (the initial point) like point A and ends at the second point (the terminal
point) like point B. This type of vector is referred to as a displacement vector in physics.
This vector can be denoted the same way we denoted vectors in the previous videos, by
using the initial and terminal point with an arrow on top as follows: vector AB, or
we can also name the vector by using a single bold face letter with an arrow on top.
Alright let’s go over the first vector operation.
Our first vector operation is a scalar multiplication, this operation consists of multiplying a vector
by a real number c this operation can be denoted as c times vector v the real number c is referred
to as a scalar note that a scalar is not a vector so we need to make sure we don’t
write an arrow un top of the variable like we do with vectors, this way we can distinguish
a scalar from a vector. As the name suggest, scalar multiplication of a vector can be geometrically
viewed as scaling a vector by a specific factor, this can either stretch or compress the original
length of a vector, it can also change the direction of a vector.
For example say you wanted to double the length of a vector v if this vector represents the
displacement of a particle between two points and if we wanted a particle to move twice
as far in the same direction as vector v. We would denote this vector as 2 times vector v.
Here the number 2 is the scalar and it’s essentially instructing us take vector v and
double the length making sure we keep the same direction in the process.
In the same manner if we want to reduce the length of vector v by half, we would denote
it as follows one half times vector v here the scalar is one half and it is instructing us take
vector v and reduce its length by a half and keeping the same direction.
Lastly, a vector can be multiplied by a negative scalar, multiplication by a negative scalar
forces the original vector to point in the opposite direction, this operation is denoted
as negative vector v, it literally turns a vector 180 degrees in the opposite direction.
In general, the scalar multiple of a vector v by a scalar c, is the vector whose length
is the absolute value of c times the length of vector v and whose direction is the same as vector v
if c > 0 in this case a positive scalar, and the direction will be opposite to vector v if c < 0
in this case a negative scalar. Note that if the scalar c = 0 or vector v equals 0, then
it follows that c time vector v equals zero. In the end real numbers essentially work like
scaling factors, this is why they are conveniently referred to as scalars.
Now let’s think about a different scenario say, that a particle moves from point A to
point B, then the particle changes direction and moves from point B to point C, with position
vector BC. The combined effect of these movements is that the particle has moved from point
A to point C, we can represent this movement by drawing a vector from point A and ending
at point C as follows. The displacement vector AC is essentially the sum of vector AB and
vector BC, we can mathematically denote this sum as follows: Vector AC equals Vector AB
plus Vector BC. In general, if we start with vectors v and
u, we can geometrically find the sum of these two vectors by first moving vector u (making
sure we do not change its magnitude or direction), so that its tail coincides with the head or
tip of vector v as follows. We can then define the sum vector v plus vector u as the vector
whose initial point coincides with the initial point of vector v and extends to the terminal
point of vector u this vector is usually referred to as the resultant vector. The resultant
vector is used extensively in fields such as physics and engineering and its interpretation
depends on the particular application it is being used for in this example the resultant
vector represents the displacement of a particle from point A to Point C.
This method of adding vectors is sometimes called the tail to head method or triangle
law. This is the general procedure to add vectors geometrically; an alternative method
of adding vectors is by just reversing the order in which you add both vectors. For example
say we wanted to add vector u plus vector v as oppose to vector v plus vector u then
we follow the same procedure we take vector v and move it so that its tail or initial
point coincides with the tip or head of vector u and define the sum vector u plus vector
v as the vector that starts from the initial point of vector u and terminates at the tip
of vector v. Notice that we obtain the same resultant vector.
In other words vector v plus vector u equals vector u plus vector v. Notice that we also
formed a parallelogram, this illustrates yet another way of adding vectors geometrically,
all we need to do is to move vector u and vector v so that they start at the same point,
then the resultant vector lies along the diagonal of the parallelogram with u and v as sides,
this technique is referred to as the parallelogram law. Either technique can be used to find
the sum of two vectors. If we wanted to add 3 or more vectors we can
essentially just apply each law multiple times. For the most part it is a lot easier to use
the triangle law when adding 3 or more vectors together. For example if we had a third vector
say vector w, and wanted to find the sum of vector v, u and w, we would first start with
vector v and move vector u so that its tail coincides with the tip of vector v, then we
would move vector w so that its tail coincides with the tip of vector u as follows. Then
the resultant vector would be the vector that starts from the initial point of vector v
and terminates at the tip of vector w. Again the order in which the vectors are added does
not matter, in the end you should obtain the same resultant vector. Now let’s move along
to vector subtraction. Computationally subtraction of vectors is
very similar to vector addition; we can geometrically find the difference of two vectors by making
use of the triangle law. Say we want to find vector v minus vector u the easiest way to
think about vector subtraction is by treating it as a vector addition in other words we want
to find vector vector v plus negative vector u. So we take vector u and align its initial
point with the tip of vector v normally this is what you would do if you were adding both
vectors, in the case of vector subtraction we want to take the negative of this vector,
recall that a negative scalar flips the vector in the opposite direction, so we go ahead
and change the direction of vector u as follows. Then we can define vector v plus negative
vector u as the vector whose initial point coincides with the initial point of vector
v and extends to the terminal point of negative vector u, again similar to vector addition.
We can also use the parallelogram law to find the difference, all we need to do is to make
sure we change the direction of vector u so that it faces in the opposite direction then
we align both vectors so that they start at the same initial point, the resultant vector
will lie along the diagonal of the parallelogram with negative u and v as sides.
A final way to think about vector subtraction is by aligning both original vectors so that
they start at the same initial point, than we can use the fact that if we were to take
vector u and add it with vector v minus vector u we should obtain vector v.
This way we can use the triangle law to draw vector
v minus vector u in this case the resultant vector would be drawn as follows.
Personally, the first method for finding the difference between two vectors makes the most
sense but you are free to use any of these 3 methods when geometrically finding the difference
between two vectors. Also keep in mind that vector subtraction
is just a special case of vector addition. If you were to have 3 or more vectors and
are asked to carry out a combination of addition and subtraction operations with them you treat
the entire problem as if it were a vector addition problem by using the triangle law
and flipping the direction of any vector with a negative sign. Alright, in our next video
we will go over various examples that make use of vector operations.