Angle between 2 Lines - Three Dimensional Geometry, Class 12, Math Video

angle between two lines lit l1 and l2 be
two lines passing through origin with
Direction ratios a 1 B 1 C 1 and a 2 B 2
C 2 respectively let P be any point on
l1 and Q be any point on l2 let theta be
the angle between two directed lines or
P and Q now these directed line segments
are nothing but the vectors with the
components a 1 B 1 C 1 and a 2 B 2 C 2
respectively therefore the angle between
the vectors o P and O Q R cos theta is
equal to modulus of a 1 a 2 plus B 1 B 2
plus even C 2 by root over a 1 square
plus B 1 square plus C 1 square into
root over a 2 square plus B 2 square
plus C 2 square if instead of Direction
ratios Direction cosines say L 1 M 1 n 1
for l1 and l2 M 2 n 2 for l2 all given
then the angle between these two lines
are given as cos theta is equal to
modulus of L 1 M 2 plus M 1 M 2 plus N 1
n 2 past L 1 square plus M 1 square plus
N 1 squared and L 2 square plus M 2
square plus n 2 squared both are equal
to 1 angle between two straight lines in
terms of sine theta we know sine theta
is equal to root over 1 minus cos square
theta substituting the value of cos
theta we get sine theta
is equal to root over 1 minus a 1 a 2
plus B 1 B 2 plus C 1 C 2 whole square
but D 1 square plus B 1 square plus C 1
square into a 2 square plus B 2 square
plus C 2 square on simplification we get
sine theta is equal to a 1 B 2 minus a 2
B 1 whole square plus B 1 C 2 minus B 2
C 1 whole square plus C 1 a 2 minus C 2
a 1 whole square divided by root over K
1 square plus B 1 square plus C 1 square
into root over a 2 square plus B 2
square plus C 2 square condition firm
two lands with the direction ratios a 1
B 1 C 1 and a 2 B 2 C 2 to be
perpendicular if these two lines are
perpendicular then the angle between
them that is theta is equal to 90
degrees then cos 90 is equal to 0 so we
get a 1 a 2 plus B 1 B 2 plus C 1 C 2 is
equal to 0 condition for the two lines
to be parallel if two lines are parallel
then angle between them is 0
substituting theta is equal to 0 in the
formula sine theta we get the condition
for two lines to be parallel as a1 / a2
is equal to b1 / b2 is equal to c1 over
c2 now let us fire
the angle between two lines when their
equations are given if theta is acute
angle between the lines our vector is
equal to K one vector plus lambda B 1
vector and our vector is equal to theta
vector plus mu into B 2 vector then the
angle between these two straight lines
this cos theta is equal to modulus of B
1 dot v2 over not B 1 into mod B 2