MATH 251 – LECTURE 2
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 11.1–3
webAssign: 11.1–2, due 1/25 11:55 p.m.
Next week: 11.4–7
webAssign: 11.3–6, opens 1/25 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.
Vectors
Definition 1 (Geometric definition). A vector (in R3) is an arrow with a length and a direction.
Definition 2 (Algebraic definition). A vector (in R3) is an ordered triple a = ha1, a2, a3i.
Vectors
A representation of a vector a = ha1, a2, a3i is a directed line segments from a point A(x, y, z) to a point
B(x + a1, y + a2, z + a3).
Conversely, given two points A(x1, y1, z1) and B(x2, y2, z2) we can form the vector AB = hx2 − x1, y2 − y1, z2 −
z1i.
Vectors
The position vector of a point A = (a1, a2, a3) is the vector a = ha1, a2, a3i.
Vectors
The magnitude (or length) of a vector a is the length of a (any) representation of a.
Vectors
Let a = ha1, a2, a3i and b = hb1, b2, b3i. Then, we define the vector sum to be
a + b = ha1 + b1, a2 + b2, a3 + b3i.
Let a = ha1, a2, a3i be a vector and let c ∈ R be a scalar. We define scalar multiplication by
c a = hc a1, c a2, c a3i.
Vectors
Definition 3. Two vectors a and b are said to be parallel if there exists a (non-zero) scalar c such that
a = c b.
Vectors
Properties of vectors: Let a, b, c be vectors, and k and m be scalars. Then
(1) a + b = b + a
(2) a + (b + c) = (a + b) + c
(3) a + 0 = a
(4) a + (−a) = 0
(5) k(a + b) = ka + kb
(6) (k + m)a = ka + ma
(7) (km)a = k(ma)
(8) 1a = a
Vectors
Let i = h1, 0, 0i, j = h0, 1, 0i and k = h0, 0, 1i.
Vectors
Let a = ha1, a2, a3i be any non-zero vector. Compute the length of the vector
a
.
u=
|a|
Dot product
Let a and b be two non-zero vectors, and let θ denote their intermediate angle. The dot product (or scalar
produc) of a and b is defined as
a · b = |a| |b| cos(θ).
Dot product
From the geometric interpretation we deduce that (a + b) · c = a · c + b · c.
Dot 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
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
Exercise 4. Let a = h1, 0, 2i and b = h1, −2, 0i. Compute the intermediate angel θ of a and b.