Hg
Name: Class 12 STICK TO YOUR WALL IN STUDY AREA
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
fb.me/thehgclasses linkedin.com/company/the-hg-classes instagram.com/the_hg_classes g.page/the-hg-classes-gurugram thehgclasses.co.in
3-D GEOMETRY
Direction cosines and Direction ratios of a line:
Direction cosines of a line (l, m, n) are the cosine of the angles made
by the line with positive directions of the coordinate axes.
l
2
+ m
2
+ n
2
= 1
For a line joining two points P(x
1
, y
1
, z
1
) and Q(x
2
, y
2
, z
2
) the direction
cosines are
l =
, m =
, n =
.
Direction ratios of a line are numbers which are proportional to the direction cosines of a line.
If l, m, n are the direction cosines and a, b and c are the direction
ratios of a line then:
l =
; m =
;
n =
Skew Lines:
• Those lines in space which are neither parallel nor intersecting. They lie on different planes.
• Angle b/w skew lines is the angle b/w 2 intersecting lines drawn from a point (preferably origin) || to
each of the skew lines.
Equations:
Passing through the given point whose PV is
, and || to a given vector
=
+
Passing through two points whose PVs are
and
.
=
+ (
-
Passing through a point (x
1
, y
1
, z
1
) and having direction cosines l, m, n
=
Passing through two points (x
1
, y
1
, z
1
) and (x
2
, y
2
, z
2
)
=
Angle b/w two lines:
If lines are given in vector form =
+
, =
+
cos θ =
If lines are given in cartesian form where
l
1
, m
1
, n
1
and l
2
, m
2
, n
2
are direction cosines
OR
a
1
, b
1
, c
1
and a
2
, b
2
, c
2
are direction ratios of the two lines respectively.
cos θ = |l
1
l
2
+ m
1
m
2
+ n
1
n
2
|
=
Distance b/w lines:
Shortest distance b/w two skew lines is length of the line segment to both lines.
If lines are given in vector form =
+
, =
+
If lines are || then:
Coplanar lines:
Condition for two lines =
+
, =
+
to be coplanar
= 0
Hg
Name: Class 12 STICK TO YOUR WALL IN STUDY AREA
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
fb.me/thehgclasses linkedin.com/company/the-hg-classes instagram.com/the_hg_classes g.page/the-hg-classes-gurugram thehgclasses.co.in
Plane:
Various Equations:
At a distance d from the origin and is the unit vector normal to the
plane through the origin.
. = d
Passing through a point whose PV is
and to the vector
-
).
= 0
Passing through 3 non collinear points whose PVs are
( −
) . [(
−
) × ( −
)] = 0
At a distance of d from the origin and the direction cosines of normal to
the plane as l, m, n
lx + my + nz = d
Passing through a point (x
1
, y
1
, z
1
) and to a given line whose direction
ratios are A, B, C
A(x-x
1
) + B(y-y
1
) + C(z-z
1
) = 0
Passing through 3 non-collinear points (x
1
, y
1
, z
1
), (x
2
, y
2
, z
2
) and(x
3
, y
3
, z
3
)
= 0
Intercept form
x/a + y/b + z/c = 1
Family of planes passing through the intersection of 2 given planes
.
= d
1
and .
= d
2
OR
A
1
x + B
1
y + C
1
z + D
1
= 0 and A
2
x + B
2
y + C
2
z + D
2
= 0
.
- d
1
) + .
- d
2
) = 0
(A
1
x + B
1
y + C
1
z + D
1
) +
(A
2
x + B
2
y + C
2
z + D
2
) = 0.
Distance of a point from a plane:
Vector form:
|d -
.|
Cartesian form:
