Parametric Equations for the line that passes through the point
P(x1, y1,z1) and is parallel to the vector
:
x = x1 + at
y = y1 + bt
z = z1 + ct
Find parametric equations for the line that passes through
P1(4,−1, 3) and is parallel to
Does the point Q(6,0,−1) lie on the line?
Does the point R(1,2,−3) lie on the line?
Is the line parallel to the vector
?
Is the line parallel to the vector
?
Find parametric equations for the line that passes through
the points P(2, 3,−2) and Q(−3,0,5).
Are there other possibilities?
Examine the line: x = 4 − t, y = 2, z = t
1. Find two vectors parallel to the line.
2. Find two points on the line.
3. Sketch the line.
Intersecting Lines
Examine the two lines:
L1 : x1 = 3+ 5t, y1 = 2 − 5t, z1 = 10t
L2 : x2 = 2 + s, y2 = −7 + 3s, z2 = 8 − 2s
Are they parallel?
Do they intersect?
What about……
L1 : x1 = 3+ 5t, y1 = 2 − 5t, z1 = 10t
L2 : x2 = 2 + s, y2= −7 + 3s, z2 = 4 − 2s
Are they parallel?
Do they intersect?
Planes in 3-D
Given: P1(x1, y1, z1) is a specific point on the plane
is a vector perpendicular to the plane
Find: A linear equation to describe all of the points P(x, y, z) on
the plane
Linear Equation for the plane that passes through the point
P1(x1, y1,z1)
and is perpendicular (normal) to the vector
a(x − x1) + b(y − y1) + c(z − z1) = 0
Find the equation for the plane that passes through P1(4,−1, 3)
and has normal vector
.
Does the point Q(6,0,−1) lie on the plane?
Does the point R(−2,−13,−3) lie on the plane?
Find the equation for the plane that contains the points
P(1,1,6) Q(2,−3,1) R(4,1,−2)
Two ways to write the equation of a plane:
1. To find the equation when you know information about the
plane, use the dot product form :
2. To find information about the plane when you know the
equation, use the general linear form:
Find the equation of the plane that contains the point
P(1,2, 3) and is parallel to the plane 2x − 4y + z = 10.
Find parametric equations for the line that contains the point
P(1,2, 3) and is perpendicular(normal) to the plane
2x − 4y + z = 10.
Find the equation of the plane that contains the point
P(1,2, 3) and is perpendicular to the line with parametric
equations x = 4 − t, y = 2 + 2t, z = 5 + t .
Find the distance from the point P(1,2, 3) to the plane
4x − 7y + 2z = 5
