Properties of Cross Product Video Lecture | Mathematics (Maths) Class 12 - JEE

remember once we started the properties
of dot product this time is to study the
properties of cross product or vector
product so the first property is non
commutativity that means definitely your
a cross B is not equal to B cross a this
is non commutativity in case of vector
product why because if I have to speak
about a cross B that will be e b
sine-theta
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in one cap a unit vector and in the
second case of B cross a things will
change to another unit vector in two cap
so in the first case your vector a B and
n one unit vector will form a right
handed system but here B a and n 2 will
form a right handed system so the order
changes and hence the commutativity is
not followed so it is non commutative
the second is stating the properties
when scalars are introduced so m is a
scalar so M a cross B can also be solved
like first you draw a draw the system
with a cross B you solve it and then
multiply the scalar or you can take em
with B first and then cross product
accordingly
so it is like that but when two scalars
are introduced one is M another is n
then what happens individually MN can be
separated then firstly M can be
separated n can be treated with me and
so on so this is just simplifying your
calculation whatever suits you best you
can use it so introduction of scalars in
the case of vector product is this
property all about the third property as
you can see a cross B plus C B plus C
cross a this is what you have done so
this is regarding distributive it t so
distributivity is followed hence a cross
B will be treated first then this is a
plus symbol then a cross C
and this is known as your left
distributivity hence I have written left
distributive mean distributive it means
LD and sorry for that fumble
now here B cross a introducing plus sign
in between then is secrecy so this is
your right distributivity writing Rd for
show
so this says distributivity is followed
in case of vector product or cross
product let's move to the next property
wherein we do not have plus but minus
this will also be satisfied so it
becomes a cross B minus of a cross C
last but not the least is this property
stating when a cross product is given to
you as zero what does it imply if a
cross product or a vector product is
zero it directly states that the two
vectors must be parallel for the two
vectors actually be nonzero this can be
stated in more detail see a cross B is
given to you as zero that means a cross
B is equal to zero and it is nothing but
a b sine-theta is given to you as zero
this is zero vector now a B is there and
since I have removed vector sign I can
even ignore this unit vector because it
states direction I am ignoring this now
these both are given to you as what as
nonzero so these both cannot be 0 so
sine theta is zero that means theta is
equal to zero degree or 180 degree that
means pipe so these are parallel because
of the fact that the angle is either
zero degree or it is PI so Phi means
anti parallel but yes these properties
are of wide importance especially you
have to take care that scalar product is
commutative but vector product is not
the rest other properties remain one or
less the same but yes the placement of
sine here and cause there makes a huge
difference