Scalar (or Dot) Product of 2 Vectors Video Lecture - Mathematics (Maths) Class 12 - JEE

scalar product of two vectors in this
module you will learn to find the scalar
product of two vectors the scalar
product of nonzero vectors a and B is
defined as the product of magnitudes of
vector a vector B and cosine of a
smaller angle between them dot is the
operation symbol for the scalar product
the scalar product of vectors a and B
can also be interpreted geometrically if
a perpendicular is drawn from the
endpoint of vector a onto vector B then
o n represents the orthogonal projection
of vector a on vector B now consider the
right triangle o am the ratio of the
base to the hypotenuse that is magnitude
of vector a is equal to cos theta this
equality helps us to deduce that the
base of this right triangle is magnitude
of the vector a into cos theta
thus the scalar product of vectors a and
B is the product of magnitude of vector
B and the scalar component of vector a
along vector B
similarly dot product of the vector B
and vector a can also be obtained by
multiplying magnitude of vector a and
the scalar component of vector B along
vector a the dot product of vector a and
B can be obtained by either multiplying
the projection of vector a on vector B
by magnitude of vector B or by
multiplying the projection of vector B
on vector a by magnitude of vector a
this fact helps us to conclude that the
dot product of two vectors is
commutative the dot product of two
vectors is maximum when theta is equal
to zero as cos theta will have its
maximum value one when theta is equal to
zero this fact helps us to conclude the
dot product of unit vector I cap with
itself is equal to one similarly J cap
dot J cap is equal to 1 and K cap dot K
cap is also equal to 1
the dot product of two vectors is
minimum for theta equal to PI as cos
theta has minimum value that is minus 1
for theta equal to PI this fact helps us
to conclude that dot product of two
vectors is negative of the product of
their magnitudes but theta equal to PI
next if a and B are two vectors and
lambda is any scalar then the dot
product of lambda times vector a and
vector B is equal to lambda into dot
product of vectors a and B which is
further equal to dot product of vector a
and lambda times vector B in this module
you have learned that the scalar product
of nonzero vectors a and B having angle
theta between them is given by magnitude
of vector a into magnitude of vector B
into cos theta where theta is the
measure of small angle between vectors a
and B when dot product of vectors a and
V is given the angle theta between the
vectors a and B is given by cos inverse
of dot product of vectors a and B upon
magnitude of vector a into magnitude of
vector B
dot product of two vectors is
commutative the dot product of two
vectors is the product of the magnitudes
patita
is equal to zero dot product of unit
vector i cap with itself j cap with
itself and k cap with itself is equal to
1 the dot product of two vectors is
equal to 0 when theta is equal to PI by
2 dot product of unit vectors that is
ICAP dot J cap J cap dot k cap and k cap
dot i cap is equal to 0 dot product of
two vectors is negative of the product
of their magnitudes when the angle
between the two vectors is equal to pi
dot product of vector a with the sum of
vectors B and C is equal to dot product
of vectors a and B plus the dot product
of vectors a and C dot product of lambda
times vector a with vector B is equal to
lambda into dot product of vectors a and
B which is further equal to dot product
of vector a and lambda times vector B