Lecture 1 (Text Sections 12.1 and 12.2)
Vectors
A vector is a directed line segment and it is represented
graphically as an arrow in space with the direction of the vec-
tor given by the direction of the arrow and the magnitude of
the vector given by the length of the line segment.
A vector has an initial point A and a te→minal point B and
is often represented symbolically in the form AB.
Graphically vectors can be added, subtracted, and multi-
plied by a scalar. For our purposes, an analytic formulation of a vector is re-
3 2
quired in R (R ). This formulation is obtained by subtract-
ing the coordinates of its initial point, A = (0 ;y0;z0) from
the coordinates of its terminal→point, B = (1 ;1 ;1 ), with the
resulting vector represented as AB= ⟨x 1 x ;0 −1y ;z 0 z1⟩: 0
→
The magnitude or norm or length of the vector AB, repre-
→
sented symbolically as | AB |, is given by the distance between
its initial and terminal point using the formula
→
| AB | =
Example: Find the representation and length of a vector
with initial point (1;−2;3) and terminal point (4;0;−3). Example ▯nd the length of the vector v = ⟨3;−7;1⟩:
Vectors expressed with respect to a coordinate system may
be added, subtracted, and multiplied by a scalar using the
following rules:
v = ⟨x ;1 ;1 ⟩1 w = ⟨x ;y 2z ⟩2 c2∈ R
i. v + w =
ii. v − w =
iii. cv =
Example: If v = ⟨3;−7;1⟩ and w = ⟨−2;0;3⟩,
▯nd 2v − 3w. Properties of Vector Operations
The following properties hold for vectors u, v, and w and
scalars a and c.
1. u + v =
2. (u + v) + w =
3. v + 0 =
4. v + (−v) =
5. c(u + v) =
6. (a + c)v =
7. 0v =
8. c0 =
9. 1v =
10. a(cv) = A unit vector is a vector of length one. If v is not the zero
vector, then a unit vector in the direction of v is given by
^ =
Example: Find the unit vector in the direction of the vector
w = ⟨−2;0;3⟩.
The standard unit vectors are given by: i = e1= ⟨1;0;0⟩,
j = ^e2= ⟨0;1;0⟩, and k = ^ e3= ⟨0;0;1⟩. These vectors
point in the direction of the positive x, y, and z coordinate
axes respectively.
Any vector in R can be represented as a sum of scalar mul-
tiples of the standard unit vectors:
⟨a;b;c⟩ =.

More
Less