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## PowerPoint Slideshow about ' Analytic Geometry of Space Second Lecture ' - landen

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Contents

- Orthogonal Projection
- Direction Cosines of a line
- 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

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 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 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 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)/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]

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

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 cos2 + y cos2, + z cos2

3. Angle Between Two Directed lines

- Because OP’ = OR’P1’P’ so we have

r cos = xcos2 + ycos2 + zcos2

- Because x=r cos1,y = r cos1 and z = rcos1

We have

cos = cos1cos2 + cos1cos2 + cos1cos2

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

cos = +

3. Angle Between Two Directed Lines

- From the last equation

cos = +

it result some implication

1. Two lines are parallel if

1 = 21 = 21 = 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 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

- 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!

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