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Hi everyone, welcome to this series of videos
for the topic of vectors.

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This chapter is important for mathematics
and it also forms the basis for the next chapter,

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that is 3-Dimensional Geometry.

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There will be series of videos for this topic.

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In this first video, I will just like to talk
about very basic stuff.

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I will try to define vectors.

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I believe that there is a lot of confusion
among students about even basic things like

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position vector, line vector, unit vector
and magnitude.

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I think if you can really focus on this video,
and try to understand the different between

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position vector, line vector — that will
clarify a lot of things for you.

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Let me start by defining the position vector.

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First, before I go into position vector, let
me just briefly summarize vectors for you

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in case you are not familiar.

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I believe you have already learned about vectors
in chapter of physics like kinematics, forces.

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You should know that vectors are quantities
that have both magnitude and direction.

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And they are very important for scientific
studies in general.

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Let me start by defining position vector.

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So position vector, is a vector connecting
origin to a point.

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And as the names suggest, basically this position
vector is about position of one point.

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I think this is where a lot of students get
confused.

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A good thing about vectors is that they will
be always be in 3 dimensions and they will

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be really helpful in the next chapter of 3
dimensional geometry.

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It is important for you to understand position
vectors and like line vector, unit vector.

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So let us just initially focus on position
vector.

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So what is the definition: a position vector
is a vector connecting origin to a point.

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Let us say we have a point P with coordinates
Px Py Pz.

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Let us call this x axis, y axis and z axis.

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Position vector is a vector connecting origin,
which is O to the point P, and pointing in

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direction on P.

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This direction is very important because if
it is in the opposite direction, then the

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vector will be opposite.

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When we write OP vector (this is called OP
vector), the first letter is from where the

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arrow line connects to second point (or the
second alphabet, which is P in this case).

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OP vector also signifies the direction of
the vector.

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What I am trying to say is that if you have
OP vector then the vector will start from

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O and end at P.

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In this case, OP vector means Px \i_cap .. (I
hope you are familiar with \i_cap: \i_cap

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means the direction of x-axis) .. + Py \j_cap
+ Pz \k_cap.

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So x axis, y axis and z axis.

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Another point you might want to note is that
magnitude of OP vector is (Px^2+Py^2+Pz^2)^(1/2).

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Let us do a quick example.

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If we have a point
A with coordinates (3,2,1).

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What is position vector and its magnitude?

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And the answer is: OA vector = 3 \i_cap +
2 \j_cap + \k_cap and OA vector magnitude

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= (3^2+2^2+1^2)^(1/2) = 14^(1.2).

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Position vector is always for a point.

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Please do not forget this basic definition
of position vector.

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Trust me that it will happen sometimes during
the chapter of vector that you will get confused

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if you forget this definition.

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So please try to remember that position vector
is always and always for a point.

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Now let us try to move to new type of vectors
called the line vector.

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Position vector is for a point.

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Line vector is for a line.

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As simple as that.

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If you have a line, let us say connecting
two points A and B.

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If I write a vector here and b vector here,
what does this mean?

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a vector is for point A and b vector is for
point B.

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Whenever you see something like this in brackets,
it means it is for a point, or it is a position

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vector.

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In other words, a vector is OA vector and
b vector is OB vector.

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And OA vectors and OB vector are position
vectors of point A and B.

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Even this basic definition gets people really
confused.

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So whenever you see a point and something
written in the bracket, that always means

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a position vector.

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So what does line vector mean?

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Line vector means connecting two points.

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If we have AB vector: starting from A and
ending at B.

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Arrow goes from A to B.

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This vector is defined as b vector - a vector.

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Or in other words, AB = OB - OA.

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Let us try to take one quick example to help
us understand this concept.

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What is the vector
joining \sin(\theta) \i_cap + \cos(\theta)

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\j_cap and (0,0,1).

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What is its magnitude?

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Let us try to find the answer.

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It might be little confusing because of the
way it is written here.

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\sin(\theta) \i_cap + \cos(\theta) \j_cap
is basically a point for which you have been

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given a position vector.

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\theta can be any parameter basically — \theta
is an angle and a parameter here.

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We have not been given a value but we can
still calculate the answer.

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b vector is \k_cap.

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What will be b - a? It will be, \k_cap - \sin(\theta)
\i_cap - \cos(\theta) \j_cap.

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And thus magnitude of AB vector would be similar
to the way we defined vector magnitude: squaring

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all the components i.e. ( -\sin(\theta) ) ^2
+ ( - \cos (\theta) )^2 + 1^2 )^(1.2). And

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\sin^2(\theta) + \cos^2(\theta) = 1. So AB
vector magnitude is 2^(1/2).

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In this part I just wanted to emphasize that
line vector is for a line and position vector

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is for a point.

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Now there are also vectors which people get
confused about.

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Here, both the vectors were fixed i.e. the
vectors started from A and went to B.

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This started from O and went to P.

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However, you should realize that sometimes
vectors are not fixed. They are only fixed in the

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magnitude and the direction but aren’t fixed
in the position.

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For instance, if I have been given a vector
a and I have been told that it is \i_cap +

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\j_cap + \k_cap and I have not been given
any other information about this.

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What would this vector represent? This vector
could represent a lot of things.

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For instance, it can represent a point such
that OA vector for a point (1,1,1) would be

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\i_cap + \j_cap + \k_cap.

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Similarly, this can also represent a line
vector which it can be (b_vector - a_vector).

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It can join two points and the difference
can come as \i_cap + \j_cap + \k_cap.

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By free vector I just know the direction and
magnitude. It is free to translate in 3-D

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space.

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Let us say this is the vector. If this vector
freely translates — it is not changing its

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direction and it is not changing its magnitude.

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If you have not been specified whether a vector
starts from origin or a position, or it joins

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two points, it can also be a free vector.

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I want to emphasize this you will see later
in this chapter that many times people get

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confused about free vectors as they are not
able to understand it. They also get confused

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about the vector connecting two points, and
when they see notation — a_vector in the

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brackets and b_vector in the brackets.

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Just to summarize: whenever you see a point,
it always means a position vector.

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Whenever you see a vector joining two points,
it means a line vector which can be calculated

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like this.

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Whenever you see something just written like
this and not specified whether it is a point

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or whether it is a line vector, that means
it is a free vector which only has a fix magnitude

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and a direction and can move freely around
the space.

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Till now I have only talked about the magnitude
of the vector here: you can just take x_component

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^ 2 + y_component^2 + z_component^3 and take
a square root of that.

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However, there is something that is also important
for a vector and that is its direction.

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So let me start by defining unit vector.

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Whenever I write a vector_a, I can also write
vector_a as magnitude of vector_a into unit

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vector.

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What does this mean?

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This means that a vector has some magnitude
which you can calculate by squaring x_component,

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y_component, z_component, and adding them
and taking a square root. And this a_cap is

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nothing but a unit vector.

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Something which I really prefer to use is
to call it “direction”.

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Unit vector and direction are synonymous.

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So whenever you see a vector: it is multiplication
of its magnitude and its direction (or its

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unit vectors).

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For instance, if we have to calculate unit
vector of a_vector. Then you can write a_cap

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as vector_a / | a |

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Vector_a in this case is \i_cap + \j_cap +
\k_cap.

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And | a | = (1^2+1^2+1^2)^(1/2), and this
means (\i_cap + j_cap + \k_cap)/3^(1/2).

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In other words I can also write that a_vector
= 3^(1/2) ( \i_cap + \j_cap + \k_cap )/3^(1/2).

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This is the direction, this is the unit vector
and this is the magnitude.

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Let us write another question.

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What is the direction of x-axis?

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Direction means a unit vector. So in that
case a_cap would be direction or something

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with the magnitude of 1. We know the direction
of x-axis is \i_cap and it also has a magnitude

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of 1. So it is just \i_cap. \i_cap is nothing
but a direction.

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So whenever I write Px \i_cap + Py \j_cap
+ Pz \k_cap, it means I have moved Px in the

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x direction, and then I have moved Py in the
y direction, and then I have moved Pz in the

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k direction.

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That is how you define direction by unit vectors.

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In this video we covered very basic stuff.

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Position vector refers to a point.

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Line vector refers to vector joining two points.

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Free vectors refers to a vector which is neither
a point nor a line, and something that can

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move freely around the space though it has
a fixed magnitude and fixed direction.

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Unit vector means the direction where you
divide the vector by its magnitude. We did

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a quick calculation to see that.

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I hope this was clear. Just take some time
and digest these things. And in the next video

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we will talk about addition of vectors and
section formula. Thank you.