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Rubono Setiawan, M.Sc . Analytic Geometry of Space Second Lecture . Contents. Orthogonal Projection Direction Cosines of a line Angle Between Two Directed Lines. 1. Orthogonal Projection.

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contents
Contents
  • Orthogonal Projection
  • Direction Cosines of a line
  • Angle Between Two Directed Lines
1 orthogonal projection
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 projection2
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 projection3
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 cosine s of a line

z

P3



C

P2

P

B

y

O

A

P1

x

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 angles

are the direction cosines

of the line

l= cos  = x/|OP|

m=cos  = y/|OP|

n=cos  = z/|OP|

2 direction cosine s of a line1
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 arecalled 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)
2 direction cosine s of a line2
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

2 direction components of the line t h rough two points
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 t h rough two points1
2.Direction Components of the line Through two Points
  • The direction cosines of the line P1P2 are

l=cos  = |P1L|/d= (x2-x1)/d

m= cos  =|P1M|/d= (y2-y1)/d

n =cos  = |P1N|/d =(z2-z1)/d

  • Hence, a set of direction component of the line joiningP1the points (x1, y1, z1) andP2 (x2, y2, z2) is [x2-x1, y2-y1, z2-z1]
3 angle between two directed lines

z

2 : 2, 2, 2

1 : 1, 1, 1

P

O

y

R

P1

x

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

3 angle between two directed lines1

z

2 : 2, 2, 2

1 : 1, 1, 1

P

O

y

R

P1

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

3 angle between two directed lines2
3. Angle Between Two Directed lines
  • Because OP’ = OR’P1’P’ so we have

r cos = xcos2 + ycos2 + zcos2

  • Because x=r cos1,y = r cos1 and z = rcos1

We have

cos = cos1cos2 + cos­1cos2 + cos1cos2

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

cos = +

3 angle between two directed lines3
3. Angle Between Two Directed Lines
  • From the last equation

cos = +

it result some implication

1. 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

3 angle between two directed lines4
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 :
  • Theorem

Two 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 1
4. Set Of Problems - 1
  • Show that the quadriliteral with vertices (5,1,1), (3,1,0), (4,3,-2), and (6,3,-1) is a rectangle
  • Find the area of the triangle with the given points A(2,2,-1), B(3,1,2) and C(4,2,-2)
  • What is known about the direction of a line if a.) cos α = 0 b.) cos α=0 and cos β=0

c.) cos α = 1.

  • Find the direction cosines of a line which makes equal angles with the coordinate axes.
  • A line has direction cosines l =cos = 3/10, m = cos= 2/5. What angle does it make with z-axis? If thisline pass through the origin give a point that passed through by this line and sketch it!
4 set of problems 2
4. Set Of Problems-2
  • Find the angle between two lines whose direction component are

and

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