Math 251 Section 11.4
Lines and Planes
Lines
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Vector Equation of a Line
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r ( t )  r0  t v
If the initial point of each vector is the
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origin and v is parallel to the line, then the terminal point of each vector, r (t ) , is a
point on the line. There are infinitely many ways to write a vector equation of a line,
just as a two dimensional line has infinitely many point slope forms.
Ex 1.
Find a vector equation of the line through the points P(2, 1, 6) and Q(-2, -1, 3).
Parametric Equations of a Line
x  x 0  at
y  y 0  bt
z  z 0  ct
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Note that this line written in vector form is r  x 0 , y 0 , z 0  t a , b, c
The numbers a, b, c are called the direction numbers of the line.
Ex 2.
a) Write parametric equations for the line of Ex 1.
b) Write parametric equations for the line through P(1, 4, -5) and Q(3, 7, -1).
Symmetric Equations of a Line
x  x0
a

y  y0
b

z  z0
c
provided a, b, c are
nonzero.
Ex 3. a) Given P(1, -1, 2) and Q(3, 4, -6), write equations for the line through P and Q
all three ways, in vector form, in parametric form and in symmetric form.
b) Find the intersection of this line with the x,y-plane.
Ex 4. Given two lines, L1 and L2 , either find their intersection point or show they
are parallel or show they are skew.
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L1 is the line through P(2, -1, 4) and Q(1, 3, -1).
L2 is given by r  t 2,  1,6 .
Planes
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For a given poin t, P ( a , b , c ), and a given vector, n , the set of points Q ( x , y , z )
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for which n  PQ  0 defines a plane.
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The vector, n , is called a normal vector to the plane.
Ex 5. Find an equation of the plane through O(0, 0, 0) which is orthogonal to
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n =  1,3,5 .
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Ex 6. Find an equation of the plane through P(2,-3,5) with normal vector n = 4,6,3
The general equation of a plane is Ax  By  Cz  D  0
Ex 7. Find a normal vector to the plane, 2 x  4 y  5z  10 .
Ex 8. Find the distance from the point Q(1, 0, -2) to the plane 3x  4 y  7 z  12 .
Ex 9. Find the line of intersection of the planes 4 x  3 y  2 z  9 and 5 x  3 y  z  0 .