Three Dimensional Geometry

= 3 D Geometry =

Axes
There are three axes in 3-D Geometry, X , Y & Z. There are 8 quadrants in 3d axes.

x-coordinate is the distance of the point from y-z plane.

Division of a line lying in a plane of axes

Suppose the XZ plane is dividing a line, it does so in the ratio -y1/y2.

Similarly for yz and xy planes ,too.

Direction Cosines
If α β γ are the angles made by a line with positive direction of X, Y and Z axes, then they are called as direction angles of the line.

The cos of α, β & γ are called as direction cosines of the line.

cos2α + cos2β + cos2γ = 1 

If l, m , n are direction cosines of a given line , and a,b,c are any real numbers such that :-

l/a = m/b = n/c ; then a,b,c are called as direction ratios of a line.

If P(x1,y1,z1) and Q(x2,y2,z2) then the direction Ratios of PQ are :-

x2- x1, y2 - y1 , z2 - z1

If AB = ax + by + cz, then direction ratios of line AB are a, b ,c.

Suppose Ray BD bisects angle ABC, then the direction ratios of Bisector BD are equal to sum of the Direction ratios of AB and BC

The vector li + mj + nk is a unit vector always. while ai + bj + ck is not a unit vector.

Angle between two Lines
If a1 b1c1 and a2b2c2are the direction ratios of two given lines, then

cos θ = a1.a2/ |a1||a2|

or cos θ = l1l2 + mm2 + n1n2

Area of Triangle
Area of triangle = 1/2 AB x AC

= Lines =

Vector Equation of a Line
1) If a line passes through a point and is parallel to another vector, then r = a + λb

2) In non-parametric form, rxa = rxb

Cartesian Equation
x - x1 / a = y - y1 / b = z - z1 / c {Derived from 2 point form }

In 2 point form :- x - x1 / x1 - x2 = y - y1 / y1- y2 = z - z1 / z1 - z2

a = x1 - x2 ; b = y1- y2  ; c = z1 - z2

Note :

For a point on the line, the Cartesian equation can be used to find the point coordinates. (equate the cartesian equation with a constant )

x = x1 + ka

y = y2 + kb

z = z2 + kc

Where, k is a constant that decides the distance of a point from the point A , on the line.

Collinearity of 3 points
three points are collinear if  a1/a2 = b1/b2 = c1/c2

Section and Mid-point Formula
For Section formula you can either consider m : n or λ : 1

Skew Lines
Shortest Distance between 2 lines in plane is given by :-

[a2 - a1 b1   b2] / |b1 x b2|

For intersecting lines, [a2 - a1 b1   b2] = 0 ... Shortest distance = 0

= Plane =

Equation of a Plane
r.n = a.n

r.n = p

For plane passing through intersection of two planes ;

a + λb = 0 (Where a & b are the equations of the two planes)

Angle between planes
cosθ = n1.n2/ |n1||n2|

Between line and plane :

sinθ = n.b / |n||b|

Co-planarity of two lines
If angle between a line and plane is 0, it lies in the plane. (This can be used to check coplanarity )

Distance between a Point & Plane
l(AP) = (a.n - p) / n

l(AP) = ( ax + by + cz + d ) / sqrt (a2 + b2 + c2)

where (x,y,z) = coordinates of point in space.

ax + by + cz + d is equation of plane.

Tips and Tricks

 * 1) In formulas dealing with ratios (e.g. cartesian equation of line, condition for parallel and perpendicular lines) direction ratios can be replaced with direction cosines.