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Analytic Geometry of Space Second Lecture

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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Analytic Geometry of Space Second Lecture

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

  2. Contents • Orthogonal Projection • Direction Cosines of a line • Angle Between Two Directed Lines

  3. 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

  4. 1. Orthogonal Projection • Example

  5. 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

  6. 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

  7. 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|

  8. 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)

  9. 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

  10. 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)

  11. 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]

  12. 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

  13. 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

  14. 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 = +

  15. 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

  16. 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

  17. 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!

  18. 4. Set Of Problems-2 • Find the angle between two lines whose direction component are and

  19. 4. Set Of Problems - 3

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