Scalar Dot product - Vector Algebra, Math, Class 12 Video Lecture

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Taylor or dot product so if I have two
nonzero vectors a and B the dot product
is denoted by e dot B so we have this is
nothing but actually a dot B is nothing
but magnitude of a in the mine shall be
into cos theta where theta is angle
between e and E so now I can take this
theta also or you can take this it also
doesn't matter because this guy will be
Magnum 180 minus theta and cos of 180
minus it is cos theta but then we will
take the smallest angle correct so you
have two vectors a and B you want to
find the dot product that will be a B
cos T let's derive this how we got a B
cos theta I let's suppose the same log I
have and apply the force on this apply
the force on there is what you see is
that this guy moves right and the whole
journey I have applied the force so in
this case in this plus the force I have
applied and let's assume that my
displacement was in this direction like
V is V and let's assume theta is the
angle between them correct so if you
observe carefully by this I can break
this force into 2 pi right so this guy
will be F cos theta this direction will
have F cos theta component if you take
this guy into this part F cos theta and
this content will be F sine T correct so
if you see if you compare the total work
total work is nothing but force into X
election into displacement in X
Direction plus force in Y displacement
in y correct force in X was F cos theta
displacement in X was the actual
displacement so that is you can see that
let's suppose my D vector plus force in
Y was F sine theta and displacement in y
was 0 because the object moved from here
to here there was no movement of object
from a to a this direction right so this
is zero this guy is zero total work done
if you see is nothing but F into D into
cos theta correct total dumb term that
came out to be was F into D into cos
theta and thus the dot product they told
was nothing but a into B into cos theta
head force talking about the magnitude
of force I'm talking about the magnitude
of this displacement so it is F into D
into cos theta correct and that was one
of the derivation of dot product using
the example of this it's the reverse way
of finding the dot product Y this F cos
theta some people ask me why it is f cos
theta Y it is not F a sorry a V cos
theta Y does not a be tan theta y is not
Y it is not a B cot theta Y is always a
V cos theta so I generally take an
example of the force and the
displacement with that you can pull and
he found that the this component
actually the the component where we have
displacement example in this case a then
you just find the component of B in this
direction right this becomes this
direction B vector will be B cos theta
and you multiply this you get a B cos
theta that is what a dot product all
about you you put both the
the vector in the same direction and
then you multiply so you get a and the
square becomes because either you
multiply this you get a B cos theta
right if you are taking this as the
frame of reference you you find the
component of a in this this direction B
direction this becomes e cos theta and
this B the same direction then you just
multiply and you get a scalar product
that's all it is so let's have some
observation the observation first
audition is the the scalar product is a
real number that is it gives a scalar
output a dot B if a and B is nonzero
vector then a dot B 0 if and only if
they are perpendicular either
perpendicular does the only case because
we had told that neither a is equal to 0
not B is equal to 0 a and B has some
value example 5 or 6 any in number but
if a dot B is 0 that is possible if and
only if a in their company because we
told that a dot B is nothing but a V cos
theta if theta is 90 this becomes
correct and the whole thing is observe
because if a and B have non zeros the
only possibility where a dot B is 0 is
is perpendicular to similarly if theta
is 0 then a dot B is a dot B itself that
means if I have a vector if I B vector
in the same direction a dot B you just
multiply these two if I have a vector
like this and B vector perpendicular the
dot product will be 0 because logically
for this B vector if you find the cost
component there how much impact this B
vector s is having on this direction
this is 0 impact is B vectors having
so the product will be 0 if you take
this case where this is a and this is B
the last Commerce in parrot is B vector
is having on a direction we say full
impact because both are in the same
direction
all right T dash equal to minus PI a dot
V is minus AV here or you see if I a
here and B here buta its other way round
right in this direction B then you will
say how much impact B is having on this
direction is a negative impact and
negative full impact actually that's why
a dot B will be minus 9 inch to a money
go that's why I can see that when you
talk about I dot I or J dot J or K dot k
you get 1 because I dot I is nothing but
they are all in the same direction J or
J see indirection K dot k situation but
you say I dot J or J or K or K dot I you
get zero because they are perpendicular
and they are parallel luckily since they
have perpendiculars right I dot J is
equal to 0 K is going to kid or die here
T putting anything you multiply any
Cross thing I or J I got K anything you
get zip this is the scalar
multiplication
please note then you see I dot hi you
get 1 J dot J is also 1 K dot here or
someone but and say I dot J that becomes
on Z dot K also becomes sir
scalar product is commutative a dot B is
equal to B dotting anyway they are
scalar so you say a dot B or say V dot a
you will get the same answer but this
will not be true for the next kind of
product vector product I will show you
that
so now we'll learn how to find the angle
between the vector using scalar product
I know that
vector a dot vector B is equal to
magnitude of and a manager V into cos
theta with this I can say cos theta is
going to divide both the side by a
magnitude of a magnitude of B magnitude
of a man if it'll be this gets cancelled
so you get cos theta is going to make
that a into vector B by mansion of a in
dementia B or theta is going to cause
inverse of this one correct its take
some property of scalar product it says
that if a and B are a two vectors if you
have two vectors a and B and lambda a
any scalar then if you say lambda into a
dot B or you say lambda into a dot B or
you a lambda B or you say this way also
this is same acne so you know this part
so you you multiply the way you want
doesn't matter it will be the same value
you have this lambda K you multiply
first and then you see dot product or
you say lambda B then you have a dot
product or you say lambda into a dot B
or you have say lambda into V dot a
everything gives the same result this is
one the property of scalar product
now let's the write the scalar product
it's to the derivation you we say that a
dot B is going to a because he died
let's see how it is so we know that if
this in this case is a 1 I plus a 2 J
plus K 3 K and B is B 1 I plus B 2 J
plus B 3 K in this case a dot B vector
that is vector a dot vector B a scalar
and that value is a 1 B 1 plus a 2 B 2
plus a TV and this guy is killable
please note there is no I or J it's a
scalar thing
vector this guy is a vector this is a
vector and is escape this is what the
formula is and it's derived this to
write this I just multiply a dot B so
that is nothing but even I plus a 2 J
plus K 3 K dot B even I Class B 2 J plus
B 3 K correct it's your dot product this
is nothing but a 1 B 1 into I dot I that
is a 1 B 1 I into 5 this is 1 then you
multiply a one with this B 2 this
becomes a 1 B 2 by a dot J correct this
is a scalar this is a vector
multiplication actually plus even
multiplied with B 3 K this becomes a 1 B
3 I dot k correct plus 9 take a to
multiply a 2 with B 1 it will be 1 I
into sorry J into I J into I correct
plus a 2 with B 2 it will be 2 J dot J
correct plus a 2 will be 3 it will be 3
J dot k j dot k plus it is gone a 3 is
running a3 b2 b1 a3 b1 k dot i a3 b2
head or j plus a3 b3 K dot k here not
now we know that we already know that a
light here I into I is equal to 1 J into
J Jade
J is 1 K dot k is also 1 but I dot J I
dot K J dot J or J dot I you multiply
any of these this is 0 correct so what
will happen I don't I will become month
this becomes even be even even be even
plus I dot J 0 this becomes 0 I dot K 0
this becomes 0
Giada is 0 this becomes 0 J dot J is 1
so this becomes a to be 2 J dot K is 0 0
0 right 0 J or J 0 K dot k 1 so class
easy and that is my answer and this is a
scalar number there is no I or J or K
Conference is not a vector is a scalar
number so if I multiply vector a and B
such an a is a 1 I plus a 2 J plus K 3 K
and B is even I plus B 2 J plus B 3 k
you do a dot product what you get is a
scalar quantity and this collar is a 1 B
1 plus a 2 B 2 plus it really and it
says that's the derivation to prove
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