Unit 9 Lesson 1
Pre-calculus Honors
Unit 9 Lesson 1: Vectors in a Coordinate Plane
Objective: _________________________________________________________________
Do Now: Read and markup the following information about vectors.
A vector is an object that shows both magnitude and direction.
Representing vectors geometrically with a directed line segment.
Initial Point
Terminal Point
Vectors are often denoted by boldface letters such as v, w, u.
The magnitude of the vector v is the length of the vector. We denote magnitude by v .
Two vectors are equal if they have the same direction and magnitude. This means that if we take
vector and translate it to a new position (without rotating it), then the vector we obtain at the end
of this process is the same vector we had in the beginning.
Different ways to represent a vector:
 directed line segment (stating initial and terminal points)
 standard position: the directed line segment whose initial point is the origin
You can change a directed line segment into standard position (keeping the magnitude and directions
the same) by writing a vector in component form: with initial point P(p1, p2) and terminal point Q (q1,
q1 - p1, q2 - p2 = v1, v2 = v
q2), then
Unit 9 Lesson 1
Reflection Questions: Answer the questions below about the reading
1.
How can you calculate the magnitude of a vector in the coordinate plane?
2.
How can you calculate the direction of a vector in the coordinate plane?
3.
What is the difference between a directed line segment and a vector in standard position?
4.
What is the purpose of component form?
Guided Practice: Representing Directed Line Segments
Verbal
Algebraic
Graphical
Find the component form
Directions: Find the
Directions: Graph the directed line segment of
and magnitude of the
component form and
vector v and the component form of vector v on
vector v that has initial
magnitude of vector v.
the same coordinate plane below.
point (4, -7) and the
terminal point (-1, 5).
Unit 9 Lesson 1
Group Practice: Vector Operations
Directions: Read and mark up the geometric definitions of vector operations and
algebraic definitions of vector operations. Use the definitions to complete the link
sheet.
Geometric Definition of Vector Scalar Multiplication Verbal
Geometrically, the product of a vector v and a
scalar k is the vector is k times as long as v.
Let v = <-2, 5> and w = <3,4>.
Note: a scalar is a number, not a vector.
If k is positive the kv is the same direction as v
and if k is negative, kv is the opposite direction of
v.
Geometrically
Directions: Represent 3v geometrically and -1/2w
geometrically. v and w are illustrated on the graph
below.
Algebraically
Definitions of Vector Addition and Scalar
Multiplication
Let u = < u1, u2 > and v = < v1, v2 > be vectors.
Let c be a scalar.
1. u + v = < u1 + v1, u2 + v2 >
2. cu = < cu1, cu2 >
3. u – v = < u1 - v1, u2 - v2 >
A) Find 3v algebraically.
B) Find -½ w algebraically
Unit 9 Lesson 1
Geometric Definition of Vector Addition
To add two vectors geometrically, position
them without changing their length or direction
so that the initial point of one coincides with
the terminal point of the other.
Verbal
Let v = <-2, 5> and w = <3,4>.
The sum of u + v is formed by joining the initial
point of the second vector v with the terminal
point of the first vector u. This technique is
called the parallelogram law for vector addition
because u + v is the diagonal of a parallelogram
having u and v as its adjacent sides.
Geometrically
Directions: Represent v + 2w geometrically.
Let v = <-2, 5> and w = <3,4>.
Algebraically
Definitions of Vector Addition and Scalar
Multiplication
Let u = < u1, u2 > and v = < v1, v2 > be vectors.
Let c be a scalar.
1. u + v = < u1 + v1, u2 + v2 >
2. cu = < cu1, cu2 >
3. u – v = < u1 - v1, u2 - v2 >
Directions: Find v + 2w algebraically.
Unit 9 Lesson 1
Geometric Definition of Vector Subtraction
To represent u – v geometrically, you can
reverse the direction of the vector you want to
subtract, then add the two vectors like the
previous example.
Verbal
Let v = <-2, 5> and w = <3,4>.
u - v is equal to u + (-v).
Geometrically
Directions: Represent 2w-v geometrically.
Let v = <-2, 5> and w = <3,4>.
Algebraically
Definitions of Vector Addition and Scalar
Multiplication
Let u = < u1, u2 > and v = < v1, v2 > be vectors.
Let c be a scalar.
1. u + v = < u1 + v1, u2 + v2 >
2. cu = < cu1, cu2 >
3. u – v = < u1 - v1, u2 - v2 >
Directions: Find 2w-v algebraically.
Pre-Calculus Honors Homework PG 511 #(1, 3, 5, 9, 11, 13-20)
Unit 9 Lesson 1
Answer Key
#1.
#3.
equivalent <3, -2>
equivalent <-2, -2>
#5.
<5, 2>
Magnitude = 29
#9.
<-2, -24>
Magnitude = 24
#11. <-11, -7>
#13.
#14.
#15.
#16.
#17.
#18.
#19.
#20.
<1, 7>
<-3, -1>
<-3, 8>
<6, 12>
<4, -9>
<-10, -10>
<-4, -18>
<-1, -7>