Section 3.3
73
Vectors in the Plane
Course Number
Section 3.3 Vectors in the Plane
Instructor
Objective: In this lesson you learned how to write the component forms
of vectors, perform basic vector operations, and find the
direction angles of vectors.
Important Vocabulary
Date
Define each term or concept.
Vector v in the plane The set of all directed line segments that are equivalent to
given directed line segment PQ, written v = PQ.
Standard position The representative of a set of equivalent directed line segments
whose initial point is the origin.
Zero vector A vector whose initial point and terminal point both lie at the origin;
denoted by 0 = ⟨0, 0⟩.
Unit vector A vector v such that ||v|| = 1.
Standard unit vectors The unit vectors ⟨1, 0⟩ and ⟨0, 1⟩, denoted by i = ⟨1, 0⟩ and
j = ⟨0, 1⟩, which can be used to represent any vector v = ⟨v1, v2⟩.
Direction angle The angle (measured counterclockwise) that a unit vector makes with
the positive x-axis.
I. Introduction (Page 291)
A directed line segment has an
terminal point
and a
initial point
What you should learn
How to represent vectors
as directed line segments
.
The magnitude of the directed line segment PQ, denoted by
||PQ||
, is its
length
. The magnitude of a
directed line segment can be found by . . .
using the
distance formula.
II. Component Form of a Vector (Page 292)
A vector whose initial point is at the origin (0, 0) can be uniquely
represented by the coordinates of its terminal point (v1, v2). This
is the
component form of a vector v
v = v1 , v 2 , where v1 and v2 are the
, written
components
of v.
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.
Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE
Copyright © Houghton Mifflin Company. All rights reserved.
What you should learn
How to write the
component forms of
vectors
74
Chapter 3
Additional Topics in Trigonometry
The magnitude (or length) of v is:
(q1 − p1)2 + (q2 − p2)2
||v|| =
=
v12 + v22
Example 1: Find the component form and magnitude of the
vector v that has (1, 7) as its initial point and (4, 3)
as its terminal point.
v = ⟨3, − 4⟩; ||v|| = 5
III. Vector Operations (Pages 293−295)
In operations with vectors, numbers are usually referred to as
. Geometrically, the product of a vector v and
scalars
a scalar k is . . .
the vector that is |k| times as long as the
What you should learn
How to perform basic
vector operations and
represent them
graphically
vector v.
If k is positive, kv has the
k is negative, kv has the
same
opposite
To add two vectors geometrically, . . .
direction as v, and if
direction.
position them
(without changing length or direction) so that the initial point of
one coincides with the terminal point of the other.
This technique is called the
parallelogram law
for
vector addition because the vector u + v, often called the
resultant
of vector addition, is . . .
the diagonal
of a parallelogram having u and v as its adjacent sides.
Let u = ⟨u1, u2⟩ and v = ⟨v1, v2⟩ be vectors and let k be a scalar
(a real number). Then the sum of u and v is the vector:
u+v=
⟨u1 + v1, u2 + v2⟩
and the scalar multiple of k times u is the vector:
ku =
k⟨u1, u2⟩ = ⟨ku1, ku2⟩
Example 2: Let u = ⟨1, 6⟩ and v = ⟨− 4, 2⟩. Find:
(a) 3u
(b) u + v
(a) ⟨3, 18⟩
(b) ⟨− 3, 8⟩
Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE
Copyright © Houghton Mifflin Company. All rights reserved.
Section 3.3
75
Vectors in the Plane
Let u, v, and w be vectors and c and d be scalars. Complete the
following properties of vector addition and scalar multiplication:
1. u + v =
v+u
2. (u + v) + w =
3. u + 0 =
u + (v + w)
u
4. u + (− u) =
0
5. c(du) =
(cd)u
6. (c + d)u =
cu + du
7. c(u + v) =
cu + cv
8. 1(u) =
u
9. 0(u) =
0
10. ||cv|| =
|c| ||v||
IV. Unit Vectors (Pages 295−296)
To find a unit vector u that has the same direction as a given
nonzero vector v, . . .
divide v by its magnitude, that is:
u = v/||v||.
In this case, the vector u is called a
direction of v
.
unit vector in the
Example 3: Find a unit vector in the direction of v = ⟨− 8, 6⟩.
v/||v|| = ⟨− 8/10, 6/10⟩ = ⟨− 0.8, 0.6⟩
Let v = ⟨v1, v2⟩. Then the standard unit vectors can be used to
represent v as v =
called the
v2 is called the
v1i + v2j
, where the scalar v1 is
horizontal component of v
vertical component of v
sum v1i + v2j is called a
linear combination
and the scalar
. The vector
of the
vectors i and j.
Example 4: Let v = ⟨− 5, 3⟩. Write v as a linear combination of
the standard unit vectors i and j.
v = − 5i + 3j
Example 5: Let v = 3i − 4j and w = 2i + 9j. Find v + w.
v + w = 5i + 5j
Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE
Copyright © Houghton Mifflin Company. All rights reserved.
What you should learn
How to write vectors as
linear combinations of
unit vectors
76
Chapter 3
V. Direction Angles (Page 297)
If u is a unit vector and θ is its direction angle, the terminal point
Additional Topics in Trigonometry
What you should learn
How to find the direction
angles of vectors
of u lies on the unit circle and
u = ⟨x, y⟩ =
⟨cos θ, sin θ⟩
(cos θ)i + (sin θ)j
=
Now, if v is any vector that makes an angle θ with the positive
x-axis, it has the same direction as u and
v=
||v|| ⟨cos θ, sin θ⟩
=
||v|| (cosθ)i + ||v|| (sin θ)j
If v can be written as v = ai + bj, then the direction angle θ for v
can be determined from tan θ =
b/a
.
Example 6: Let v = − 4i + 5j. Find the direction angle for v.
128.66°
VI. Applications of Vectors (Pages 298−299)
Describe several real-life applications of vectors.
Answers will vary.
What you should learn
How to use vectors to
model and solve real-life
problems
Homework Assignment
Page(s)
Exercises
Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE
Copyright © Houghton Mifflin Company. All rights reserved.