Properties of Scalar Product and Projection of a Vector on a Line Video Lecture | Mathematics (Maths) Class 12 - JEE

projection of a vector on a line
you
let us consider a vector PQ which makes
an angle theta with a directed line L in
anti-clockwise direction the projection
of vector PQ on line L is a vector P
whose magnitude is magnitude of PQ
vector into cos theta and the direction
is either positive or negative depending
on angle theta for theta greater than
zero and less than 90 degree cos theta
is positive so vector P has the same
direction as line L
for theta greater than 90 degree and
less than 180 degree cos theta is
negative so the direction of vector P is
opposite to direction of line L the
magnitude of P will remain same as
magnitude of R cos theta the vector P
along line L is called the projection
vector and the magnitude of P vector is
called as the projection of vector PQ
along line L
if pcap is the unit vector along a line
L then the projection of a vector a on
the line L is given by vector a dot P
cap
projection of factor e on any other
vector b is given by dot product of
vector e and unit vector P cap unit
vector in direction of P vector is B by
magnitude of V vector so projection is
equal to scalar product of a dot B by
magnitude of B vector
if alpha beta and gamma are Direction
angle of a vector a is equal to a 1 I
cap plus a 2 J cap plus a 3 K cap then
it's Direction cosines may be given as
cos alpha is equal to a dot I cap by
magnitude of a vector or cos alpha is
equal to a 1 by magnitude of a vector
similarly cos beta is equal to a 2 by
magnitude of a vector and cos gamma is
equal to a 3 by magnitude of a vector or
we can rewrite them as a 1 is equal to
magnitude of a vector cos alpha a 2 is
equal to magnitude of a vector cos beta
a 3 is equal to magnitude of a vector
cos gamma magnitude mod of a vector cos
alpha is the projection of vector a
along x axis or a 1 is the projection of
vector a along x axis
similarly a2 and a3 are projection of
vector a along y-axis and z-axis
what