MATH 251 – LECTURE 3
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.
Dot product (or Scalar product)
We defined a · b = |a| |b| cos(θ).
We’ve deduced that (a + b) · c = a · c + b · c and, similarly, c · (a + b) = c · a + c · b.
Let a = ha1, a2, a3i and b = hb1, b2, b3i. Compute a · b without first computing θ.
Dot product (or Scalar product)
Exercise 1. Let a = h1, 0, 2i and b = h1, −2, 0i. Compute the intermediate angel θ of a and b.
Cross product (or Vector product)
The Vector product a × b is the unique vector such that
(1) a × b is orthogonal to both a and b. (That is, a · c = b · c = 0.)
(2) |a × b| is equal to the area of the parallelogram spanned by a and b. (That is, |a × b| = |a||b| sin(θ).)
(3) a, b, a × b fulfill the right hand rule.
Cross product (or Vector product)
Exercise 2. h1, 0, 0i × h0, 1, 0i =
Exercise 3. h0, 0, 1i × h0, 1, 0i =
Cross product (or Vector product)
Properties of the Cross product:
(1) a × b = −b × a
(2) (ka) × b = k(a × b) = a × (kb)
(3) a × (b + c) = a × b + a × c
(4) (a + b) × c = a × c + b × c.
In coordinates:
a × b = h a2b3 − a3b2, a3b1 − a1b3,
a2 a3 a1 a3 , −
= b1 b3 ,
b2 b3 Exercise 4. h2, 3, 1i × h−1, 1, 1i =
a1b2 − a2b1 i
a1 a2 b1 b2 Cross product
Exercise 5. Find the area of the triangle with vertices A(1, 1, 1), B(0, 0, 0), and C(2, −1, 4).
Cross product
Exercise 6. Find the volume of the parallelepiped spanned by the vectors h1, 2, 3i, h4, 2, −3i and h0, −2, 0i.
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 7. Find the symmetric equations of the line L passing through h1, 2, −3i of direction h−1, −1, −1i.
Exercise 8. Find the symmetric equations of the line L passing through h4, 1, 0i of direction h0, −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
Any two non-parallel planes H1 and H2 intersect in a line L.
Exercise 9. Determine a parametric representation of the intersection of the two planes defined by
2x + 3y + 2z = 0 and 4x − y − z = 0.
Exercise 10. Determine the symmetric equations of the intersection of the two planes defined by
x + y − z = 8 and 4x − 2y + z = 2.
Exercises
Exercise 11. Determine the distance from the point A(2, 4, 5) to the plane defined by
x + 2y + 3z = 0