VECTORS IN THE PLANE
Section 10.2a
Vectors in the Plane
Some quantities only have magnitude, and are called scalars
… Examples?
Some quantities have both magnitude and direction, and can
be represented by directed line segments… Examples?
The directed line segment AB has initial point A and
terminal point B; its length is denoted by AB . Directed
line segments that have the same length and direction are
equivalent.
Terminal
point
AB
Initial
B
point
A
Vectors in the Plane
Definitions: Vector, Equal Vectors
A vector in the plane is represented by a directed line
segment. Two vectors are equal (or the same) if they have
the same length and direction.
Definition: Component Form of a Vector
If v is a vector in the plane equal to the vector with initial
point 0, 0 and terminal point v1 , v2 , then the
component form of v is




v  v1 , v2
The magnitude (length) of v is
v  v  v2
2
1
2
Finding Component Form
Find the (a) component form and (b) length of the vector
with initial point P = (–3, 4) and terminal point Q = (–5, 2).
Start with a graph of the vector.
(a)
v  5   3 , 2  4  2, 2
(b)
v
 2    2 
2
2
 82 2
If a vector has a magnitude of 1, then it is a unit vector.
The slope of a nonvertical vector is the slope shared by the
lines parallel to the vector.
The zero vector:
direction.
0  0, 0
is the only vector with no
Vectors in the Plane
Definitions: Vector Operations
Let u  u1 , u2 , v  v1 , v2 be vectors with k a scalar
(real number).
Addition: u  v  u1  v1 , u2  v2
Subtraction: u  v  u1  v1 , u2  v2
Scalar Multiplication: ku  ku1 , ku2
Negative (opposite):  u   1 u  u1 , u2
Vectors in the Plane
Vector Addition
When adding vectors geometrically, align them “head to tail,”
and the sum is called the resultant vector. This geometric
description of vector addition is sometimes called the
parallelogram law:
u
v
u+v
v
u
Vectors in the Plane
Scalar Multiplication
When multiplying scalars and vectors geometrically, the
scalar simply stretches (k > 1) or shrinks (k < 1) the vector.
If k is negative, the vector also changes to the opposite
direction:
u
0.7u
2u
–2u
Vectors in the Plane
Properties of Vector Operations
Let u, v, and w be vectors and a, b be scalars.
1. u  v  v  u
2.  u  v   w  u   v  w 
3. u  0  u
4. u    u   0
5. 0u  0
6. 1u  u
7. a  bu    ab  u
8. a  u  v   au  av
9.  a  b  u  au  bu
Vectors in the Plane
Definition: Dot Product (Inner Product)
The dot product (or inner product) u v (“u dot v”) of
vectors u  u1 , u2 and v  v1 , v2 is the number
u v  u1v1  u2 v2
Definition: Angle Between Two Vectors
The angle between nonzero vectors u and v is


u
v
1
  cos 

u
v


Practice Problems
Find the component form of the vector v of length 3 that
makes an angle of 137 with the positive x-axis.
v  3cos137,3sin137  2.194, 2.046
Let
u  1,1 and v  2, 5 . Find:
4u  v  4 1,1  2, 5  6,9
2u  3v  2 1,1  3 2, 5  8, 17
 8   17   353
2
2
Practice Problems
Find the measure of angle C in the triangle ABC defined by
the following points: A 0, 0


B  3,5  C  5, 2 
Sketch a graph of the triangle.
The angle is formed by vectors
CA
and
CB .
Component forms of these vectors:
CA  0  5, 0  2  5, 2
CB  3  5,5  2  2,3
 CA CB 
1 

Angle between these vectors: m C  cos
 CA CB 


Practice Problems
Find the measure of angle C in the triangle ABC defined by
the following points: A 0, 0


B  3,5  C  5, 2 
CA  5, 2 CB  2,3
 CA CB 
1 

m C  cos
 CA CB 



1 
 cos



 5 2    2  3 
2
2
2
2 
 5   2   2   3 
Practice Problems
Find the measure of angle C in the triangle ABC defined by
the following points: A 0, 0


B  3,5  C  5, 2 
CA  5, 2 CB  2,3

1 
m C  cos



1 
 cos



 5 2    2  3 
2
2
2
2 
 5   2   2   3 


4
  78.111  1.363
29
13 

 
Practice Problems
Find a unit vector in the direction of the given vector.
4, 3
Since a unit vector has a magnitude of 1, simply
divide the given vector by its own magnitude
(this will keep the direction the same but stretch
or shrink the vector to the correct length).
Unit Vector:
4, 3
4, 3
4, 3

u

2
2
5
16  9
4   3
4 3
,
How can we verify this answer? 
5 5