MAT 1236
Calculus III
Section 12.5 Part I
Equations of Line and
Planes
http://myhome.spu.edu/lauw
HW…

WebAssign 12.5 Part I
(9 problems 75 min.)
Preview

Equations of Lines

Equations of Planes
• Vector Equations
• Parametric Equations
• Symmetric Equations
Recall: Position Vectors


Given any point P  a1 , a2  , OP  a1 , a2 is the
position vector of P.
To serve as a position vector, the initial
point O of the vector is fixed.
Equations of Lines

In 2D, what kind of info is required to
determine a line?
• Type 1:
• Type 2:

Q: How to extend these ideas?
Vector Equations



Ingredients
•
•
A (fixed) point P  x , y , z  on the line
A (fixed) vector 𝑣 =< 𝑎, 𝑏, 𝑐 > parallel to the
line
0
0
0
0
Any vector parallel to the line can be
represented by ________________
The position vector of a (general) point
𝑃(𝑥, 𝑦, 𝑧) on the line can be represented
by ________________
Parametric Equations
r  r0  tv
x, y, z  x0 , y0 , z0  t a, b, c
v  a, b, c
Example 1

Find a vector equation and parametric equations for
the line that passes through the point (1,1,5) and is
parallel to the vector < 1,2,1 >.
Vector Equation
r  r0  tv
Example 1
Vector Equation
r  r0  tv
Example 1
x 

y 
z 

Vector Equation
r  r0  tv
Example 1: Parametric Equation
x 

y 
z 

Can you recover (1,1,5) and < 1,2,1 >
from the parametric equation?
Remarks

As usual, parametric equations are not
unique (e.g. 𝑣1 =< −2, −4, −2 > gives
another parametric equation.)
Example 1: Symmetric Equation
x  1  t , y  1  2t , z  5  t
Example 1: Symmetric Equation
x  1  t , y  1  2t , z  5  t
Can you recover (1,1,5) and < 1,2,1 >
from the symmetric equation?
What if…
x  1  t , y  1  2t , z  5  t
If one of the component is a constant,
then…
3 Possible Scenarios

Given 2 distinct lines in 3D, they are
either
•
•
•
Example 2
Show that the 2 lines are parallel.
 L1 : x  1  t , y  1  2t , z  5  t

 L2 : x  5  2s, y  3  4s, z  2s
Example 2: Plan
Show that the 2 lines are parallel.
 L1 : x  1  t , y  1  2t , z  5  t

 L2 : x  5  2s, y  3  4s, z  2s
1. Find 𝑣1 such that 𝑣1 ||𝐿1
2. Find 𝑣2 such that 𝑣2 ||𝐿2
3. Show that 𝑣1 ||𝑣2 which implies 𝐿1 ||𝐿2
Expectations

Give precise reasons.
Incorrect Logic...
Show that the 2 lines are parallel.
 L1 : x  1  t , y  1  2t , z  5  t

 L2 : x  5  2s, y  3  4s, z  2s
Since 𝑣1 ||𝐿1 , 𝑣1 =< 1,2,1 >.
Example 3
Find the intersection point of the 2 lines
 L1 : x  2t , y  3  4t , z  1  t

 L2 : x  1  s, y  3s, z  s
s  1, t  0
 0,3,1
Example 3
Find the intersection point of the 2 lines
 L1 : x  2t , y  3  4t , z  1  t

 L2 : x  1  s, y  3s, z  s
(The lines intersect if there is a pair of
parameters (𝑠, 𝑡) that gives the same point
on the two lines.)
s  1, t  0
 0,3,1
Example 3: Plan
Find the intersection point of the 2 lines
 L1 : x  2t , y  3  4t , z  1  t

 L2 : x  1  s, y  3s, z  s
1. Set up 2 equations in 𝑠 and 𝑡
2. Solve for 𝑠 or 𝑡
3. Find the point corresponds to the 𝑠 or 𝑡
s  1, t  0
 0,3,1
Example 3
Find the intersection point of the 2 lines
 L1 : x  2t , y  3  4t , z  1  t

 L2 : x  1  s, y  3s, z  s
(Note that it is given that the two lines
intersect each other. If this is not given,
then the solutions steps will be different.)
s  1, t  0
 0,3,1
Expectations

You are expected to carefully explain
your solutions. Answers alone are not
sufficient for quizzes or exams.
Example 4
Show that the two lines are skew.
 L1 : x  1  t , y  2  3t , z  4  t

 L2 : x  2s, y  3  s, z  3  4s
Example 4: Plan
Show that the two lines are skew.
 L1 : x  1  t , y  2  3t , z  4  t

 L2 : x  2s, y  3  s, z  3  4s
(a) Show that the two lines are not parallel.
(b) Show that the two lines have no
intersection points.
13i  6 j  5k
(1), (2)  t 
11
8
,s 
5
5
Example 4: Plan for (b)
Show that the two lines are skew.
 L1 : x  1  t , y  2  3t , z  4  t

 L2 : x  2s, y  3  s, z  3  4s
13i  6 j  5k
1. Assume the lines intersect at a point
11
8
(1), (2)  t  , s 
5
5
2. Set up the 3 equations in 𝑠 and 𝑡
3. Choose 2 systems of 2 equations to solve for
(𝑠, 𝑡) and see that the system is inconsistent.
Expectations


To show that two lines are non-parallel,
you are expected to show that the cross
product of the two (direction) vectors is a
non-zero vector.
Do not substitute 𝑠 and 𝑡 directly into the
3rd equation. You are expected to
compute the values of the two sides
separately and compare the values.