MATH 251 – LECTURE 4
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 11.3–6
webAssign: 11.3–5, due 2/1 11:55 p.m.
Next week: 11.6–7, 12.1–3
webAssign: 11.6, 12.1, and 12.3, opens 2/1 12 a.m.
Help Sessions:
M W 5.30–8 p.m. in BLOC 161
Office Hours:
BLOC 641C
M 12:30–2:30 p.m.
W 2–3 p.m.
or by appointment.
The zero vector
The zero vector 0 = h0, 0, 0i has no direction!
Lines
A line L is determined by a point P0 = P0(x0, y0, z0) and a direction v. That is, the position vector r of any
point P on v can be written as
r = r0 + tv
where r0 is the position vector of P0 and t is a real parameter. This equations is called a parametric representation of the line L.
Lines
The vector equation r = r0 + tv is short for
x = x0 + tv1,
y = y0 + tv2,
z = z0 + tv3.
If v1, v2, v3 6= 0, then we obtain the symmetric equations of L:
x − x0 y − y0 z − z0
t=
=
=
.
v1
v2
v3
Exercise 1. Find the symmetric equations of the line L passing through h1, 2, −3i of direction h−1, −1, −1i.
Planes
A plane H is determined by a point P0 = P0(x0, y0, z0) and a normal vector n. That, the position vector r for
any other point P on H fulfills that
n · (r − r0) = 0.
In coordinates, with n = ha, b, ci,
ax + by + cz = n · r0 or a(x − x0) + b(y − y0) + b(z − z0) = 0.
Planes
Exercise 2. Determine the implicit equation for the plane with normal vector h1, 0, 2i passing through the point
h0, 1, 3i.
Planes
Any two non-parallel planes H1 and H2 intersect in a line L.
Exercise 3. Determine a parametric representation of the intersection of the two planes defined by
2x + 3y + 2z = 0 and 4x − y − z = 0.
Planes
Exercise 4. Determine the symmetric equations of the intersection of the two planes defined by
x + y − z = 8 and 4x − 2y + z = 2.
Exercises
Exercise 5. Determine the distance from the point A(2, 4, 5) to the plane defined by
x + 2y + 3z = 0
Quadric curves in R2
The general form of a quadratic equation in two variables is
Ax2 + By 2 + Cxy + Dx + Ey + G = 0.
Example 6. Write x2 + y 2 + 2xy + 2x + 2y = 0 in standard form.
Quadric curves in R2
Example 7. Write x2 + 2xy + 2x + 2y = 0 in standard form.
Quadric curves in R2
Each quadratic equation in two variables can be written in one of the two forms
Ax2 + By 2 + G = 0 and Ax2 + By + G = 0.
Ellipse
Hyperbola
Parabola
ax2 + by 2 − g = 0
ax2 − by 2 ± g = 0
ax2 ± ey ± g = 0