Lines and Planes

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Lecture 4

• Lines and Planes

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Lecture 4 Objectives

• Find vector, parametric, and general forms of
  equations of
• lines
• planes
• Find distances and angles between lines and
  planes

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Question What specifies a line in space?

• Answer
• Two Points,
• Or
• A Point and a vector giving the direction.

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Vector Equations of Lines in R3
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Example

• Find the vector equation of the line passing
  through the points P (1, 0, ?1) and Q (?2,
  1, 3)
• Answer First get the direction vector v
  PQ OQ ? OP
• ?2, 1, 3 ? 1, 0, ?1
• ?3, 1, 4
• Then use the formula to get
• r OP tv 1, 0, ?1 t?3, 1, 4
• So r 1 ? 3t, t, ?1 4t

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Parametric Equations of Lines in R3

• If we write the 3 component equations resulting
  from the vector equation r r0 tv, where
  r x, y, z, r0 x0, y0,
  z0, and v a, b, c
• we then get the 3 parametric equations x
  x0 at, y y0 bt, z z0 ct.

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Example

• Find the parametric equations of the line
  passing through the point P (1, 3, ?2) and
  parallel to the line with the vector equation
  r 2t, 1 t, 4 ? 3t
• Answer We put the vector equation in the form
  r OP tv 0, 1, 4 t2, 1, ? 3,
• thus getting the direction vector v 2, 1, ?
  3.
• Finally, we write the parametric equations x
  1 2t, y 3 t, z ?2 ? 3t.

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Question What specifies a plane in space?

• Answer
• 3 points A, B, and C
• Or
• a point A and two direction vectors u AB
  and v AC along the plane.

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Vector Equations of Planes in R3

• These are given by
• OP OA sAB tAC or
• r r0 su tv

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Example

• Find the vector equation of the plane passing
  through the points A (1, 0, ?1) B
  (?2, 1, 3) and C (2, ?1, 0)
• Answer First get the direction vectors u
  AB OB ? OA ?2, 1, 3 ? 1, 0, ?1
  ?3, 1, 4
• and v AC OC ? OA 2, ?1, 0 ? 1, 0, ?1
  1, ?1, 1
• Then use the formula to get
• r OA su tv 1,0,?1 s?3,1,4
  t1,?1,1
• So r 1 ? 3s t, s ? t, ?1 4s t

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Parametric Equations of Planes in R3

• If we write the 3 component equations resulting
  from the vector equation r r0 su tv,
  where r x, y, z, r0
  x0, y0, z0, u a1, b1, c1 and v a2,
  b2, c2
• we then get the 3 parametric equations x x0
  a1s a2t, y y0 b1s b2t, z
  z0 c1s c2t.

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The Normal Form of Plane Equations
Note A plane is also specified by a point and a
normal (perpendicular) vector.
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The Normal Form of Plane Equations
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Example

• Find the normal form of the equation of the
  plane through the origin (0, 0, 0) and
  perpendicular to the vector 2, 0, ?5.
• Answer 2(x ? 0) 0(y ? 0) (?5)(z ? 0) 0
  2x ? 5z 0

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Example

• Find the normal form of the equation of the
  plane through the points P (1, 0, 0) Q
  (0, 2, 0) and R (0, 0, 3).
• Answer Start with Ax By Cz D, and
  substitute the points P, Q, and R to get A
  D 2B D 3C DLetting D 1 (or any nonzero
  value), we get A 1, B 1/2, C 1/3. Thus, we
  get x (1/2)y (1/3)z 1, or
  6x 3y 2z 6

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Angles between lines (or planes)

• Definitions
• The angle between two intersecting lines is the
  angle between their direction vectors.
• The angle between two intersecting planes is the
  angle between their normal vectors.
• The angle between a line and a plane is given by
  90? ? ?, where ? is the angle between the
  direction vector of the line and the normal
  vector of the plane.

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Example

• Find the intersection point of (and the angle
  between) the line with parametric equations x
  1 ? t, y 3t, z 4 2t
• and the plane with equation 2x y 5.

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Distances between Points and Lines or Planes

• Theorem
• (The distance between a point B and a plane
  passing through a point A and perpendicular to a
  vector n) projn(AB).
• (The distance between a point B and a line
  passing through a point A and having a direction
  vector v) AB ? projv(AB).

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Example

• Show that the following two lines are parallel
  and find the distance between them x 1 ? t,
  y 3t, z 4 2t
• and x 3 ? 2t, y 4 6t, z 4t.

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• Thank you for listening.
• Wafik