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Key concepts

  • In three dimensions, the direction of a line is described by a direction vector. The vector equation of a line with direction vector v = a , b , c passing through point P = ( x 0 , y 0 , z 0 ) is r = r 0 + t v , where r 0 = x 0 , y 0 , z 0 is the position vector of point P . This equation can be rewritten to form the parametric equations of the line: x = x 0 + t a , y = y 0 + t b , and z = z 0 + t c . The line can also be described with the symmetric equations x x 0 a = y y 0 b = z z 0 c .
  • Let L be a line in space passing through point P with direction vector v . If Q is any point not on L , then the distance from Q to L is d = P Q × v v .
  • In three dimensions, two lines may be parallel but not equal, equal, intersecting, or skew.
  • Given a point P and vector n , the set of all points Q satisfying equation n · P Q = 0 forms a plane. Equation n · P Q = 0 is known as the vector equation of a plane.
  • The scalar equation of a plane containing point P = ( x 0 , y 0 , z 0 ) with normal vector n = a , b , c is a ( x x 0 ) + b ( y y 0 ) + c ( z z 0 ) = 0 . This equation can be expressed as a x + b y + c z + d = 0 , where d = a x 0 b y 0 c z 0 . This form of the equation is sometimes called the general form of the equation of a plane .
  • Suppose a plane with normal vector n passes through point Q . The distance D from the plane to point P not in the plane is given by
    D = proj n Q P = | comp n Q P | = | Q P · n | n .
  • The normal vectors of parallel planes are parallel. When two planes intersect, they form a line.
  • The measure of the angle θ between two intersecting planes can be found using the equation: cos θ = | n 1 · n 2 | n 1 n 2 , where n 1 and n 2 are normal vectors to the planes.
  • The distance D from point ( x 0 , y 0 , z 0 ) to plane a x + b y + c z + d = 0 is given by
    D = | a ( x 0 x 1 ) + b ( y 0 y 1 ) + c ( z 0 z 1 ) | a 2 + b 2 + c 2 = | a x 0 + b y 0 + c z 0 + d | a 2 + b 2 + c 2 .

Key equations

  • Vector Equation of a Line
    r = r 0 + t v
  • Parametric Equations of a Line
    x = x 0 + t a , y = y 0 + t b , and z = z 0 + t c
  • Symmetric Equations of a Line
    x x 0 a = y y 0 b = z z 0 c
  • Vector Equation of a Plane
    n · P Q = 0
  • Scalar Equation of a Plane
    a ( x x 0 ) + b ( y y 0 ) + c ( z z 0 ) = 0
  • Distance between a Plane and a Point
    d = proj n Q P = | comp n Q P | = | Q P · n | n

In the following exercises, points P and Q are given. Let L be the line passing through points P and Q .

  1. Find the vector equation of line L .
  2. Find parametric equations of line L .
  3. Find symmetric equations of line L .
  4. Find parametric equations of the line segment determined by P and Q .

P ( −3 , 5 , 9 ) , Q ( 4 , −7 , 2 )

a. r = −3 , 5 , 9 + t 7 , −12 , −7 , t ; b. x = −3 + 7 t , y = 5 12 t , z = 9 7 t , t ; c. x + 3 7 = y 5 −12 = z 9 −7 ; d. x = −3 + 7 t , y = 5 12 t , z = 9 7 t , t [ 0 , 1 ]

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P ( 4 , 0 , 5 ) , Q ( 2 , 3 , 1 )

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P ( −1 , 0 , 5 ) , Q ( 4 , 0 , 3 )

a. r = −1 , 0 , 5 + t 5 , 0 , −2 , t ; b. x = −1 + 5 t , y = 0 , z = 5 2 t , t ; c. x + 1 5 = z 5 −2 , y = 0 ; d. x = −1 + 5 t , y = 0 , z = 5 2 t , t [ 0 , 1 ]

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P ( 7 , −2 , 6 ) , Q ( −3 , 0 , 6 )

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For the following exercises, point P and vector v are given. Let L be the line passing through point P with direction v .

  1. Find parametric equations of line L .
  2. Find symmetric equations of line L .
  3. Find the intersection of the line with the xy -plane.

P ( 1 , −2 , 3 ) , v = 1 , 2 , 3

a. x = 1 + t , y = −2 + 2 t , z = 3 + 3 t , t ; b. x 1 1 = y + 2 2 = z 3 3 ; c. ( 0 , −4 , 0 )

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P ( 3 , 1 , 5 ) , v = 1 , 1 , 1

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P ( 3 , 1 , 5 ) , v = Q R , where Q ( 2 , 2 , 3 ) and R ( 3 , 2 , 3 )

a. x = 3 + t , y = 1 , z = 5 , t ; b. y = 1 , z = 5 ; c. The line does not intersect the xy -plane.

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P ( 2 , 3 , 0 ) , v = Q R , where Q ( 0 , 4 , 5 ) and R ( 0 , 4 , 6 )

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For the following exercises, line L is given.

  1. Find point P that belongs to the line and direction vector v of the line. Express v in component form.
  2. Find the distance from the origin to line L .
Practice Key Terms 9

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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