Examples : Scalar Product Video Lecture | Mathematics (Maths) Class 12 - JEE

hello friends this video on vector
algebra part 19 is brought you by exam
fear calm normal free from exam before
watching this video please make sure
that you have watched part 1 to part 18
so let's take some example they define
the angle between the two vectors a and
V that man shoots three and two and we
had told that a dot V is equal to six
so I know that a dot B vector is nothing
but magnitude of a into magnitude of V
into cost correct so a dot B is six
given to find the angle between these
two where magnitude of a and V is given
three into so magnitude of a is 3 B is 2
into cos theta 3 into 2 6 cancel so cos
theta is good 2 cos theta is good to 1
means theta is equal to 0 and that is
nice so that means both are in the same
direction so I have this vector a and
vector B both are in same direction
exactly the impression there is no angle
between them thus 3 into 2 is equal to 6
correct this guy is 3 this guy is 2 you
multiply these two you get vector AV e
dot d that is 6 in fact you've already
the vector because you multiply these
two you little scalar quantity that is 6
let's take one more example you define
the angle between the vector two vectors
so to find the angle between two vectors
what we can do is I have a I have B I
know that a dot B is nothing but
magnitude of a into magnitude of B into
cos theta correct since a and B is given
that suppose this is a this is B I can
find a dot B also I can find my image it
okay I can buy my angel B then I can
easily find cost so let's find a dot B
first a dot B is nothing but
i minus 2j plus 3k dot 3i minus 2 J plus
K I into y is 1 3 into 1 is 3 and other
will be 0
I know so I will not take that minus 2
into minus 2 is 4 plus 4 3 into 1 is 3
plus that is so I dot a dot B ayah next
I define magnitude of a magnitude of a
is nothing but root of 1 square plus 2
square plus 3 square that is 1 square
plus sorry minus 2 square plus 3 square
1 minus 2 & 3 this is nothing but root
of 1 plus 4 plus 9 that is root footing
let's find magnitude of B manager of B
will be root of 3 square plus minus 2
square plus 1 square 3 square plus minus
2 square plus 1 square that's again 9
plus 4 plus 1 that's again boon 14
correct so I have magnitude of a minus
Toby and I on a dot B also so let us put
everything in this equation a dot B is
10 is equal to magnitude of a moves 14
manage to be beautiful be into cos theta
so this if you saw cos theta is 10 by 14
that is 5 by 7 so I get theta is nothing
but cos inverse Phi I said and the
difference so finding is much more
difficult because I knew vector a and B
I knew how to find a dot B I can find my
Shero way I can find my nature to be and
I can find cos theta sorry let's take
some examples on projection we had to
find the projection of vector I minus J
on vector I plus so let us assume this
guy is vector a and this guy is like
that correct so I have this vector a and
I on this vector B I have to find
projection of a on B right
so magnitude of this B what angle is
Theta let's suppose - oh this will be
in this direction a cause theory correct
and the ration will be B vector by
modeling this is nothing but a dot B
vector by management
please note projection of a on B is a
dot B by D enter correct so here I've
defined projection of a on D this guy is
a this guy's B so in first kind of
pushing just first foot which is a
vector which is the vector that makes a
life simple then we can say that
position of a on V B is a dot B by
magnitude of B so we have to just find a
dot B a dot B will be this guy I minus J
dot I plus G divided by magnitude of B
that is magnitude of this that is root
of 1 square plus 1 square cut because
magnitude of B is nothing but root of 1
square plus conspan so this is the
projection vector right so I'll write
the projection is this guy so this is
nothing but I minus this guy is I I into
I is 1 and minus 1 into minus 1 is again
minus 1 by root 2 that is 0 so it is 0
vector actually this is a vector 0
vector correct so in this case when you
project this guy on this guy you get a 0
vector if the question would have been
to find the projection of vector this
guy on this they may have tail a dot B
by mod of a I hope you understand this
projection of a on B projection of a on
B is not equal to projection of B on you
both are different the magnitude both
are different so when you the direction
is different national
but in this case and Maggie should also
be different so in this case it will be
a dot B by B management and in this it
will be a dot B by 9 H 0 both are
different so let's take an example need
to find the projection of this vector I
plus 3 J plus 7 K on vector 7th I minus
J plus 8 so again let's say this guy is
a and this guy is B so we have to find
projection of a on V so just now we have
to write this this is nothing but a dot
B by nine it should be correct this is
my vector a B a on B let's suppose so we
know a we know B we don't know be
manageably let's find my zero B first
this becomes root of seven square seven
square plus minus 1 square plus eight
correct so this becomes root of 40 49
plus one plus h square 64 that is root
of 1 1 so i guard up Maggie all
right so let's put this in this formula
a dot B will be I plus 3j plus 7.7 I
minus G Plus H K vector by manage toggle
is without London so this becomes I into
1 into 7 is 7 plus 3 into minus 1 is
minus 3 plus 7 into minus 8 is minus 56
correct but this is a plus here actually
so 7 into it is plus the physics correct
by root 1 1 this is nothing but 56 and
plus 4 60 60 by
do one one so this guy is the prediction
of this vector on this vector and
there'll be some direction also correct
the direction of this will be same as
the direction of vector B because we are
predicting a on B and that is - thank
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