7_Lines_Planes - Lines and Planes Lines Let Let P0(a, b, c)...

Lines and Planes
Lines
Let
l
be a line parallel to vector
v
.
v = v
1
i + v
2
j + v
3
k
Let
P
0
( a , b , c )
be a point on the line and
P ( x , y , z )
be any
other point on the line.
P
0
P
is a vector parallel to the line.
P
0
P v
so
The parametric equations of a line are:
x = a + v
1
t
y = b + v
2
t
z = c + v
3
t

To find the equation of a line you need a point on the line and a
vector parallel to the line.
ex. Find the equation of the line through P(1, 5, 3) parallel to
vector
v = 2 , 0 , 1 = 2 i + k
ex. Find the equation of the line through points P(1, 2, 3) and
Q(3, 6, 9).
ex. Find a vector parallel to the line
.

x = 9 - t
y = 3
z = 6 t
ex. Determine which of the following line equations represent the
same line as
x = 9 - t y = 3 z = 6 t
x 1 = 7 + 2 t x 2 = 11 - 3 t x 3 = 9 + t
y 1 = 3 y 2 = 3 y 3 = 3
z 1 = 12 - 12 t z 2 = 10 + 18 t z 3 = 6 t
Planes

Let
P
0
( a , b , c )
be a point in the plane. If P(x, y, z) is any other
point in the plane, then
P
0
P
is a vector that lies in the plane. Let
n = n
1
i + n
2
j + n
3
k
be a vector perpendicular (normal) to the
plane. Then

The equation of a plane is
n
1
x + n
2
y + n
3
z = n ⋅ p
where
n
is the normal vector and
p
is the vector formed by the point.
To find the equation of a plane you need a point on the plane and a
vector normal (perpendicular) to the plane.