6.3 Vectors in the Plane
Many quantities in geometry and physics, such as area, time,
and temperature, can be represented by a single real number.
Other quantities, such as force and velocity, involve both
magnitude and direction and cannot be completely
characterized by a single real number.
To represent such a quantity, we use a directed line segment.
The directed line segment PQ has initial point P and terminal
point Q and we denote its magnitude (length) by PQ .
Q
Terminal Point
P
Initial Point
PQ
Vector Representation by Directed Line Segments
Let u be represented by the directed line segment from
P = (0,0) to Q = (3,2), and let v be represented by the directed
line segment from R = (1,2) to S = (4,4). Show that u = v.
S
4
v
3
Q
R
2
u
1
P
1
2
3
4
Using the distance formula, show
that u and v have the same length.
Show that their slopes are equal.
2
u =
2
(3 ! 0) + (2 ! 0)
= 13
v =
(4 ! 1)2 + (4 ! 2)2
= 13
2
Slopes of u and v are both
3
Component Form of a Vector
The component form of the vector with initial point P = (p1, p2)
and terminal point Q = (q1, q2) is
PQ
= q1 ! p1 ,q2 ! p2 = v1 , v2 = v
The magnitude (or length) of v is given by
v =
2
2
(q1 ! p1 ) + (q2 ! p2 )
=
2
v1 + v2
2
Find the component form and length of the vector v that has
initial point (4,-7) and terminal point (-1,5)
6
Let P = (4, -7) = (p1, p2) and
Q = (-1, 5) = (q1, q2).
4
Then, the components of v = v1 , v2
are given by
2
-2
-2
2
4
v1 = q1 – p1 = -1 – 4 = -5
v2 = q2 – p2 = 5 – (-7) = 12
-4
Thus, v =
-6
and the length of v is
-8
v = (!5) 2 + 12 2 = 169 = 13
! 5,12
Vector Operations
The two basic operations are scalar multiplication and vector
addition. Geometrically, the product of a vector v and a scalar
k is the vector that is k times as long as v. If k is positive,
then kv has the same direction as v, and if k is negative, then
kv has the opposite direction of v.
v
½v
2v
-v
3
! v
2
Definition of Vector Addition & Scalar Multiplication
Let u = u1 ,u 2 and v = v1 , v2 be vectors and let k be a
scalar (real number). Then the sum of u and v is
u + v = u1 + v1 , u 2 + v2
and scalar multiplication of k times u is the vector
ku = k u1 , u2 = ku1 , ku2
Vector Operations
Ex. Let v = ! 2,5 and w = 3,4 . Find the following vectors.
a. 2v
b. w – v
2v = ! 4,10
w ! v = 3 ! (!2),4 ! 5 = 5,!1
10
2v
v
8
4
6
3
4
2
2
-4
-2
-2
w
-v
1
2
-1
1
2
3
w-v
4
5
Writing a Linear Combination of Unit Vectors
Let u be the vector with initial point (2, -5) and terminal point
(-1, 3). Write u as a linear combination of the standard unit
vectors of i and j.
6
Solution
10
(-1, 3) 4
8
u = ! 1 ! 2,3 + 5
2
6
= ! 3,8
-2
2
4
u
-2
4 8j
-4
-6
-8
= !3i + 8 j
(2, -5)
Graphically,
it looks like…
2
-3i
-4
-2
-2
2
Writing a Linear Combination of Unit Vectors
Let u be the vector with initial point (2, -5) and terminal point
(-1, 3).Write u as a linear combination of the standard unit
vectors i and j.
Begin by writing the component form of the vector u.
u = "1" 2,3 " ("5)
u = "3,8
!
!
!
u = "3i + 8 j
!
Unit Vectors
v "1%
= $ 'v
u = unit vector =
v #v &
Find a unit vector in the direction of v =
v!
=
v
"2,5
("2)
2
!
+ (5)
2
1
=
"2,5 =
29
!
!
"2,5
"2 5
,
29 29
Vector Operations
Let u = -3i + 8j and let v = 2i - j. Find 2u - 3v.
2u - 3v = 2(-3i + 8j) - 3(2i - j)
= -6i + 16j - 6i + 3j
= -12i + 19 j