Shortest Distance between 2 Lines (Distance between 2 skew lines and distance between parallel lines) Video Lecture | Mathematics (Maths) Class 12 - JEE

distance between two skew lines plate l1
and l2 B to skew lines with the
equations our vector is equal to a-1
vector plus lambda B 1 vector and our
vector is equal to a2 vector plus mu B 2
vector consider any point is on the line
l1 with position vector a 1 and the
point T on the line l2 with position
vector a 2 now the magnitude of the
shortest distance vector will be equal
to that of projection of s T along the
direction of the line of shortest
distance if PQ is the shortest distance
vector between l1 and l2 then it should
be perpendicular to both B 1 and B 2
then the unit vector n vector along PQ
would be unit vector n is equal to B 1
vector cross B 2 vector divided by
modulus of B 1 vector cross B 2 vector
if D is the magnitude of the shortest
distance vector then vector PQ is equal
to D n cap let theta be the angle
between the vectors s T and P Q then P Q
is equal to s T into cos theta but the
angle between two vectors is given by
the formula cos theta is equal to
modulus of P Q vector dot s T vector
divided by modulus of P Q vector into
modulus of s T vector
substituting the corresponding values of
P Q vector and s T vector we get cos
theta is equal to D into unit vector n
into vector a 2 minus vector a 1 divided
by D into modulus of s T vector again
substituting the value of unit vector n
cap we get cos theta is equal to modulus
of B 1 cross B 2 vectors dot vector a 2
minus vector a 1 divided by s T into
modulus of vector B 1 cross vector B 2
therefore substituting the value of cos
theta in the shortest distance D is
equal to PQ which is equal to s T into
modulus of cos theta we get d is equal
to modulus of B 1 vector cross B 2
vector dot a 2 vector minus a 1 vector
the whole thing divided by modulus of B
1 vector cross B 2 vector when the lines
are given in Cartesian form then the
shortest distance between them is
modulus of determinant of X 2 minus X 1
y 2 minus y 1 Z 2 minus Z 1 a 1 B 1 C 1
a 2 B 2 C 2 divided by root over B 1 C 2
minus B 2 C 1 whole square plus C 1 a 2
minus C 2 a 1 whole square plus a 1 B 2
minus a 2 B 1 whole square