MCV4U1-UNIT EIGHT-LESSON THREE
Lesson Three: Vector, Parametric, and Symmetric Equations in  3
P
L
P1
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d  a, b, c 
Let L be a line in  3 .
Let P(x,y,.z) represent any point on L.
Let P x1 , y1 , z1  represent a specific point on L.
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Let d  a, b, c  be parallel to L.
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The vector equation of L is.... OP  OP1  P1 P
In component form, x, y, z   x1 , y1 , z1   t a, b, c
The parametric equations are.....
x  x1  ta
y  y1  tb
z  z1  tc
x  x1  ta
or
y  y1  tb
z  z1  tc
The symmetric equations are.....
x  x1 y  y1 z  z1


a
b
c
Example 1:
Answer:
Find the vector equation of the line through (1,-,1,3) and (2,2,4).
(x,y,z)=(1,-1,3)+t(1,3,1)
Example 2: Find the parametric equations of the line with y-intercept 3, and parallel to
x  7  y  1 2z  4


the line
.
4
2
6
MCV4U1-UNIT EIGHT-LESSON THREE
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d  4,2,3 and ( x1 , y1 , z1 )  0,3,0 
  x, y, z   0,3,0   t 4,2,3
 x  4t
y  3  2t
z  3t
Example 3:
Find the symmetric equations of the line parallel to the x-axis and
intersecting the z-axis at 4.

d  1,0,0  or , (2,0,0), (3,0,0), (4,0,0) ......etc.
x1 , y1 , z1   0,0,4
  x, y, z   0,0,4   t 1,0,0
x  t
y  0  0t
z  4  0t
symmetric equations don' t exit .