Analytic Geometry of SpaceSecond Lecture Rubono Setiawan, M.Sc.

Contents

1. Orthogonal Projection2. Direction Cosines of a line3. Angle Between Two Directed Lines

1. Orthogonal Projection The ortogonal projection of a point P upon any

line is defined as the foot of the perpendicular from P to the line

The projection of a line segmen P1P2 upon any line is the segment joining the projections of the endpoints P1 and P2 upon the line

The projection of a broken line upon any line is the sum of the projection of the segment forming the broken line

1. Orthogonal Projection Example

1. Orthogonal Projection

The orthogonal projection of a point on a plane is the foot of the perpendicular from the point to a plane.

The orthogonal projection on plane of a segment PQ of a line is the segment P’Q’ joining the projections P’ and Q’ of P and Q on the plane

1. Orthogonal Projection For the purpose of measuring distance and

angle, one direction along a line will be regarded as positive and the opposite direction as negative

A segment PQ on a directed line is positive or negative according as Q in the positive or negative direction from P. From this definition its follows that PQ=-QP

2. Direction Cosines of a Line Given a direct line in 3D rectangular coordinate

system. The angle , , formed by this line with the positive x-, y-, and z-axis are called direction angle.

If we make a direct line ’, parallel to trough the origin and point P (x,y,z). The direction angles of ' is also the direction angle of

The cosine of these anglesare the direction cosines

of the linel = cos = x/|OP|m = cos = y/|OP|n = cos = z/|OP| A P1

BO

P2P

P3C

z

x

y

2. Direction Cosines of a Line In fact that

|OP|= We can easily get

cos2 + cos2 + cos2 =

Consider any line (not necessarily trough the origin) whose direction cosines are proporsional to three numbers a, b, c, a:b:c = cos : cos : cos a,b, and c are called direction components of

Now the problem is How to determine direction cosine form known a, b, and c ?

We use square bracket to denote direction component as [a, b, c] to distinguish it with coordinates (x, y, z)

222 zyx

1|| 2

222

OPzyx

2. Direction Cosines of a Line Let

cos = a ; cos = b; and cos = c Find so that

cos2 + cos2 + cos2 = 1 (a2 + b2 + c2) 2 = 1 =

So we get

222

1

cba

2. Direction Components of the line Through two Points Let d is the distance between two points P1 (x1, y1, z1) and P2 (x2, y2, z2)

2. Direction Components of the line Through two Points The direction cosines of the line P1P2 are

l = cos = |P1L|/d= (x2-x1)/dm= cos =|P1M|/d= (y2-y1)/dn = cos = |P1N|/d =(z2-z1)/d

Hence, a set of direction component of the line joining P1 the points (x1, y1, z1) and P2 (x2, y2, z2) is [x2- x1, y2- y1, z2- z1]

3. Angle between Two Directed lines Let line 1 and 2 are two lines intersecting at the

origin with direction angle 1, 1, 1 and 2, 2, 2 What is ? Let P(x,y,z) a point

on 1 x = r cos 1,y = r cos 1, z = r cos 1

O

P1

P

R

1 : 1, 1, 1

2 : 2, 2, 2

z

y

x

3. Angle Between Two Directed lines

If |OP|=r, OP’ is projection segment OP upon 2 we get length of OP’ is

|OP’|=r cos In other side we can get this

OP’ by make projection of broken segment ORP1P upon 2 as OR’P1’P’|OR’P1’P’| = x cos 2 + y cos 2, + z cos 2

O

P1

P

R

1 : 1, 1, 1

2 : 2, 2, 2

z

y

x

3. Angle Between Two Directed lines

Because OP’ = OR’P1’P’ so we haver cos = x cos 2 + y cos 2 + z cos 2

Because x=r cos 1, y = r cos 1 and z = rcos 1

We havecos = cos 1 cos 2 + cos 1 cos 2 + cos 1 cos 2

If both lines are defined by direction component [a1,b1,c1] and [a2,b2,c2] we have

cos = + 22

22

22

21

21

21

212121

. cbacba

ccbbaa

3. Angle Between Two Directed Lines

From the last equation

cos = +

it result some implication1. Two lines are parallel if 1 = 2 1 = 2 1 = 2

or using direction component [a1,b1,c1] and [a2,b2,c2]

2. Two lines are perpendicular if a1a2 + b1b2 + c1c2 = 0

22

22

22

21

21

21

212121

. cbacba

ccbbaa

2

1

2

1

2

1

cc

bb

aa

3.Angle Between Two Directed Lines The condition that two given lines

are perpendicular is that cos = 0. Hence, we also have the following theorem :

TheoremTwo directed lines 1 and 2 with direction cosines l1 ,m1 ,n1 and l2 ,m2 ,n2

, respectively, are perpendicular if :l1 l2 + m1 m2 + n1 n2 = 0

4. Set Of Problems - 11. Show that the quadriliteral with vertices (5,1,1), (3,1,0), (4,3,-2), and

(6,3,-1) is a rectangle2. Find the area of the triangle with the given points A(2,2,-1), B(3,1,2)

and C(4,2,-2) 3. What is known about the direction of a line if a.) cos α = 0 b.) cos

α=0 and cos β=0c.) cos α = 1.

4. Find the direction cosines of a line which makes equal angles with the coordinate axes.

5. A line has direction cosines l =cos = 3/10, m = cos = 2/5. What angle does it make with z-axis? If this line pass through the origin give a point that passed through by this line and sketch it!

4. Set Of Problems-2

1. Find the angle between two lines whose direction component are

and

143,

148,

141

805,

801,

802

4. Set Of Problems - 3