Analytic Geometry of Space

Third Lecture

Rubono Setiawan, M.Sc.

Contents Point Dividing a Segment is a Given Ratio Another Types of Space Coordinate Systems

1.Polar Coordinates2.Cylindrical Coordinates3.Spherical Coordinates

Let a line segment with P(x1,y1,z1) and Q (x2,y2,z2), and let R(xR,yR,zR) is any point on the segment PQ so that PR:RQ = 1 : 2 = : 1

Determine coordinate of R

Point Dividing a Segment is a Given Ratio

O

P

R

Q

Q1

R1

P1

z

y

x

We get

xR = x1 + (x2-x1)/ (+1)

yR = y1 + (y2-y1)/ (+1)

zR = z1 + (z2-z1)/ (+1) Or in another form

xR = yR = zR =

If M is midpoint of PQ we have

xM = yM = zM =

Point Dividing a Segment is a Given Ratio

1

21 xx

1

21 yy

1

21 zz

221 xx

221 yy

221 zz

If is positive then R is between P and Q. If 0 > > -1 then R is outside PQ on the side of P. If < -1 then R is outside PQ on the side Q. If = -1 then r is an infinite point trough line

joining P and Q.

Point Dividing a Segment is a Given Ratio

Example: Let a line segment joining P(-2,-4,27) and Q(2,2,3). Let R(xR,yR,zR) is any point on the segment PQ so that PR:RQ = : 1 Determine R if =-0.8 Determine R if =-3

Answer: .......

Point Dividing a Segment is a Given Ratio

Set of Problems Show that the points (1,2,3), (1,3,6), (3,8,6),

(3,7,3) are the vertices of parallelogram. Then, find the coordinates of the intersection of the diagonals in these parallelogram and also its area.

Another Types of Space Coordinate System

POLAR COORDINATESLet OX, OY and OZ be a set of rectangular axes and P be any point in a space. Let OP = and have direction angle α, β, and

A

BO

P2P

C

z

x

y

The position of line OP is determined by and the position of line