3d geometry Flashcards
dist of p from
pa=\/b^2+c^2 y axis\/c^2+a^2
centroid
(x1+x2+x3/3 y1+y2+y3/3 z1+z2+z3/3)
direction cosine
cos alpha ,cos gamma,cos beta
cos
alpha=a/r beta=b/r gamma=c/r
cos^2 alpha +cos^2 beta +cos^2gamma
=1 ,l^2+m^2+n^2=1
dist b/w two points
\/(x2-x1)^2+(y2-y1)^2+(z2-z1)^2
section formula
for internal mx2+nx1/m+n for y change x to y for externally negative
ar of triangle
1/2| i j k
| x2-x1 y2-y1 z2-z1
x3-x1 //////////////
equation of line in space
x-x1/a=y-y1/b=x-x1/c
eq of line passing from 2 points
x-x1/x2-x1 =y-y1/y2-y1=z-z1/z2-z1
angle between two intersecting lines
cos@=a1a2+b1b2+c1c2/underroot (a1^2+b1^2+c1^2)(a2^2+b2^2+c2^2)
perpendicular line
a1a2+b1b2+c1c2=0
angle bw lines in term’s of direction cosines
cos-1(l1l2+m1m2+n1n2)
perpendicular line in dc
l1L2+m1m2+n1n2=0
coplanar lines
lines which lie in the same plane are called coplanar lines. Any
two coplanar lines are either parallel or intersecting
condition for coplanarity of line
for lines x-x1/l1=y-y2/m1=z-z2/n1 and same just change l2 m2 n2 if
x2-x1 y2-y1 z2-z1
l1 m1 n1 =0
l2 m2 n2
what is a plane
A plane is a surface such that line joining any two points of
the plane totally lies in it.
The general equation of a plane is
a x +by+ cz+ d = 0
Plane through the origin is
a x+ by+ cz = 0
Planes parallel to coordinate axes
by+ cz +d=0 parallel to X-axis
a x+ cz+ d=0 parallel to Y-axis
a x+ by+ d = 0 parallel to Z-axis
If l m, and n are DC’s of normal to the plane, p is the
distance of the origin from the plane, then equation of
plane is
l x + my+ nz = p.
Coordinates of foot of perpendicular, drawn from the
origin to the plane, is
(lp mp np ).
Plane through a point (x 1y1 z1) is
a(x-x1)+b(y-y1)+z(z-z1)=0
Plane ax+by+cz+d=0 intersecting a line segment
joining and divides it in the ratio
ax1+by1+cz1+d/ax2+by2+cz2+d
direction ration in terms of points
a=x2-x1 b=y2-y1 c=z2-z1
plane parallell to given plane ax+by+cz+d=o is
ax+by+cz+k=0
any plane passing through the line of intersection of plane is
(ax+by+cz+d)+k(a1x+b1y+c1z+d1)=0
plane through the origin
ax+by+cz=0
if a,b,c are intercepts
x/a+y/b+z/c=1
when p dist from origin and l m n are direction cosines
lx+my+nz=p
vector eqn of plane passing
(r-a).(b x c)=0
two plane giving eqn of straight lines
a1x+b1y+c1z+d1=0
a2x+b2y+c2z+d2=0
angle between planes
@=cos-1(a1a2+b1b2+c1c2/underoot a1^2+b1^2+c1^2 *underoot a2^2+b2^2+c2^2
dist of a pt( x y z) from plane
|ax1+by1+cz1+d | /underoot a^2+b^2c^2
distance between two parallel planes
|d1-d2|//underoot a^2+b^2+c^2
angle bw a line and a plane
sin@=|a1a2+b1b2+c1c2/underoot a1^2+b1^2+c1^2 *underoot a2^2+b2^2+c2^2
projection of line segment
|(x2-x1)L +(y2-y1)M +(z2-z1)n|
shortest or distance bw two lines
d=||(c-a) x b|/|b||
foot of the perpendicular from a point on the plane ax+by+cz+d=0
eqn=x-x1/a= y-y1/b= z-z1/c
length=a^2+b^2+c^2/underoot a^2+b^2+c^2
direction ratio of line joining two points
x2-x1/|pq| same for y and z
direction cosine in terms of direction ratios
L=+_ a/underoot a^2+b^2+c^2 for m=b for n=c
