Revision Material 31. Three Dimensional Geometry -1 Direction Cosines

JEE Syllabus:

Three dimensions: Direction cosines and direction ratios, equation of a straight line in space, equation of a plane, distance of a point from a plane.

Part 1

Coordinates, Direction cosines and direction ratios

Coordinates of a point in space

Thee mutually perpendicular lines in space define three mutually perpendicular planes which in turn divide the space into eight parts known as octants and the lines are known as the coordinate axes.

(x,y,z), the coordinates of a point P are the perpendicular distances from P on the three mutually rectangular coordinate planes YOZ, ZOX, and XOY respectively. The coordinates of a point are the distances from the origin of the feet of the perpendiculars from the point on the respective coordinate axes.

Sign convention of three dimensional geometry

All distances measured along OX, OY, and OZ will be positive.
All distances measured along or parallel to OX’, OY’, Oz’ will be negative.



x-coordinate signs and octants
Octant

OXYZ *** +
OX’YZ *** -
OXY’Z *** +
OX’Y’Z *** -
OXYZ’ *** +
OX’YZ’ *** -
OXY’Z’ *** +
OX’Y’Z’ *** -

y-coordinate signs and octants


Octant

OXYZ *** +
OX’YZ *** +
OXY’Z *** -
OX’Y’Z *** -
OXYZ’ *** +
OX’YZ’ *** +
OXY’Z’ *** -
OX’Y’Z’ *** -

z-coordinate signs and octants

Octant

OXYZ *** +
OX’YZ *** +
OXY’Z *** +
OX’Y’Z *** +
OXYZ’ *** -
OX’YZ’ *** -
OXY’Z’ *** -
OX’Y’Z’ *** -

Equation of xy plane is z = 0.
Equation of yz palne is x = 0.
Equation of xz plane is y = 0.

Distance formula

The distance between the points P(x1,y1,z1) and Q(x2,y2,z2) is given by

PQ = √[(x2-x1) ² + (y2-y1) ² + (z2-z1) ²]

Section formula

Internal division: Division of line segment PQ internally.

The coordinates of a point R that divides line segment PQ with coordinates P(x1,y1,z1) and Q(x2,y2,z2) in the ratio m1:m2 internally are



[ (m1x2+m2x1)/(m1+m2), (m1y2+m2y1)/(m1+m2), (m1z2+m2z1)/(m1+m2)]


PR:RQ = m1:m2

If the is the mid point that m1:m2 = 1:1 the coordinates are
[(x1+x2)/2, (y1+y2)/2, (z1+z2)/2]

External division: Division of line segment PQ internally.

The coordinates of a point R that divides line segment PQ with coordinates P(x1,y1,z1) and Q(x2,y2,z2) in the ratio m1:m2 externally are

[ (m1x2 -m2x1)/(m1+m2), (m1y2 -m2y1)/(m1+m2), (m1z2 -m2z1)/(m1+m2)]

PR:RQ = m1:m2


Direction cosines of a line

The direction cosines of a line: The direction cosines of a line are defined as the direction cosines of any vector whose support is the given line.

If A and B are two points on a given line L, then direction cosines of vectors AB and BA are the direction cosines of line L. If α, β, γ, are the angles which the line L makes with positive directions of x-axis, y-axis, and z-axis respectively, then its direction cosines are either cos α, cos β , cos γ or -cos α, -cos β ,- cos γ.

Therefore if l,m,n are direction cosines of a line, then –l.-m,-n are also its direction cosines.

Also l² +m² + n² = 1

If P and Q are two points with coordinates P(x1,y1,z1) and Q(x2,y2,z2) on line L, it direction cosines (of L or PQ) are

(x2-x1)/PQ, (y2-y1)/PQ, , (z2-z1)/PQ, or (x1-x2)/PQ, (y1-y2)/PQ, , (z1-z2)/PQ

Direction ratios of a line