Cheat Sheet – 3D Geometry
DOUBLEROOT
» Distance Formula
» Equation of a Plane
Distance between two points (x1, y1, z1) and (x2, y2, z2)
√(x1 − x2 )2 + (y1 − y2 )2 + (z1 − z2 )2
At a distance d from origin and DCs of normal (l, m, n)
lx + my + nz = d
At a distance d from origin and DRs of normal (a, b, c)
» Section Formula
ax + by + cz ± d√a2 + b 2 + c 2 = 0
Coordinates of the point which divides the line joining twoPassing through (x1, y1, z1) and DRs of normal (a, b, c)
points (x1, y1, z1) and (x2, y2, z2) in the ratio m:n
a(x − x1 ) + b(y − y1 ) + c(z − z1 ) = 0
mx2 + nx1 my2 + ny1 mz2 + nz1
Making
intercepts
a, b, c on the X, Y, Z axes respectively
Internally: (
,
,
)
x y z
m+n
m+n
m+n
+ + =1
mx2 − nx1 my2 − ny1 mz2 − nz1
a b c
Externally: (
,
,
)
m−n
m−n
m−n
Passing through a point (x1, y1, z1) and parallel to two
lines having DRs (a1, b1, c1) and (a2, b2, c2)
» Equation of a Line
x − x1 y − y1 z − z1
Passing through a point (x1, y1, z1) and having DRs (a, b, c)
b1
c1 | = 0
| a1
x − x1 y − y1 z − z1
a2
b2
c2
=
=
a
b
c
Passing through two points (x1, y1, z1) and (x2, y2, z2) and
Passing through two points (x1, y1, z1) and (x2, y2, z2)
parallel to a line having DRs (a, b, c)
x − x1
y − y1
z − z1
x − x1
y − y1 z − z1
=
=
x2 − x1 y2 − y1 z2 − z1
|x2 − x1 y2 − y1 z2 − z1 | = 0
a
b
c
Angle between two lines
Passing through three non-collinear points (x1, y1, z1) and
DCs (l1, m1, n1) and (l2, m2, n2)
(x2, y2, z2) and (x3, y3, z3)
cos θ = l1 l2 + m1 m2 + n1 n2
x − x1
y − y1 z − z1
DRs (a1, b1, c1) and (a2, b2, c2)
|x2 − x1 y2 − y1 z2 − z1 | = 0
a1 a2 + b1 b2 + c1 c2
x3 − x1 y3 − y1 z3 − z1
cos θ =
2
2
2
2
2
2
√a1 + b1 + c1 √a2 + b2 + c2
Angle between two planes a1x + b1y + c1z + d1 = 0 and
a2x + b2y + c2z + d2 = 0
Distance of a point (x1, y1, z1) from a line
a1 a2 + b1 b2 + c1 c2
x − x2 y − y2 z − z2
cos θ =
=
=
OR
a
b
c
√a21 + b12 + c12 √a22 + b22 + c22
x−x1
y−y1
z−z1
Distance between parallel lines
x − x1 y − y1 z − z1 x − x2 y − y2 z − z2 Angle between a line a1 = b1 = c1
=
=
&
=
=
a
b
c
a
b
c and a plane a2x + b2y + c2z + d2 = 0
î
ĵ
k̂
|x − x1 y2 − y1 z2 − z1 |
| 2
|
a
b
c
d=
√a2 + b 2 + c 2
|
|
sin θ =
a1 a2 + b1 b2 + c1 c2
√a21
+ b12 + c12 √a22 + b22 + c22
Distance of a point (x1,y1,z1) from a plane ax+by+cz=d
D = |ax1 + by1 + cz1 − d|/√a2 + b 2 + c 2
Distance between parallel planes ax + by + cz = d1
and ax + by + cz = d2
Shortest distance between skew lines
x − x1 y − y1 z − z1 x − x2 y − y2 z − z2
D = |d1 − d2 |/√a2 + b 2 + c 2
=
=
&
=
=
a1
b1
c1
a2
b2
c2 Family of planes passing through the line of
|
x2 − x1
| a1
a2
y2 − y1
b1
b2
z2 − z1
c1 |
c2
|
|√(b1 c2 − b2 c1 )2 + (c1 a2 − c2 a1 )2 + (a1 b2 − a2 b1 )2 |
intersection of the planes a1x + b1y + c1z + d1 = 0 and
a2x + b2y + c2z + d2 = 0
a1 x + b1 y + c1 z + d1 + λ(a2 x + b2 y + c2 z + d2 ) = 0
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