Ch 12.2 - Vectors
The term vector is used by scientists to indicate a quantity (such
as displacement or velocity or force) that has both magnitude and
direction.
I A vector is often represented by an arrow or a directed line
segment.
I The length of the arrow represents the magnitude of the
vector and the arrow points in the direction of the vector.
I A vector is usually denoted by printing a letter in boldface (v)
→
or by putting an arrow above the letter v .
→
I
In v =AB, A is the initial point and B is the terminal point.
I
If u =CD has the same length and the same direction as v,
then u = v and say they are equivalent (or equal).
→
Vector addition
If u and v are vectors positioned so the initial point of v is at the
terminal point of u, then the sum u + v is the vector from the
initial point of u to the terminal point of v.
I
I
The above definition is sometimes called the Triangle law.
What about the following case?
Example
Draw the sum of the vectors shown below.
Scalar Multiplication
If c is a scalar and v is a vector, then the scalar multiple cv is the
vector whose length is |c| times the length of v and whose
direction is the same as v if c > 0 and is opposite to v if c < 0. If
c = 0 or v = 0, then cv = 0.
difference u − v
Draw 2u − v
Vector components
Given the points A = (x1 , y1 , z1 ) and B = (x2 , y2 , z2 ), the vector a
→
with representation AB is
a =< x2 − x1 , y2 − y1 , z2 − z1 >
Note: If the starting point is the origin, the vector is called the
position vector.
Example) Find the vector represented by the directed line segment
with initial point A = (1, 3, −1) and terminal point B = (2, 1, −3).
Length of a vector
The magnitude or length of the vector v is the length of any of its
representations and is denoted by the symbol |v | or ||v ||.
Given a three dimensional vector a =< a1 , a2 , a3 >, its magnitude
is
|a| =
p
a1 2 + a2 2 + a3 2
Example) Find |v| if v =< 1, 2, −3 >.
Algebra of vectors in R3
Given a =< a1 , a2 , a3 > and b =< b1 , b2 , b3 >, and c a scalar,
I a + b =< a1 + b1 , a2 + b2 , a3 + b3 >
I a − b =< a1 − b1 , a2 − b2 , a3 − b3 >
I ca =< ca1 , ca2 , ca3 >
Example) Let a =< 3, 0, −1 > and b =< −1, 1, 2 >. Find |a|, and
vectors a + b, a − b, 4a, 2a + 3b.
Properties of vectors
Note: V3 denotes the set of all three-dimensional vectors.
Standard basis vectors in V3
i =< 1, 0, 0 >,
j =< 0, 1, 0 >,
k =< 0, 0, 1 >
What are the lengths of the vectors i, j, k?
I If a =< a1 , a2 , a3 >, then
a = a1 i + a2 j + a3 k
Example) Write < 2, −1, 4 > in terms of i, j, k.
Unit vectors
A unit vector is a vector whose length is 1. So, if a 6= 0, then the
unit vector u of a is
a
u=
|a|
Example) Find the unit vectors of the following
1. < 2, 1, −1 >
2. 3i − j + 2k
Example
Find a vector that has the same direction as < 6, 6, 4 > but has
length 6.
Application
A 100-lb weight hangs from two wires as shown in the figure
below. Find the tensions (forces) T1 and T2 in both wires and the
magnitudes of the tensions.
Class Exercise
If v lies in the first quadrant and makes an angle π/6 with the
positive x-axis and |v| = 2, find v in component form. Find its unit
vector.