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. Use your
markup to an answer the guided text questions.
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,
q2), then
q1 - p1, q2 - p2 = v1, v2 = v
Unit 9 Lesson 1
Guided Text Questions: Answer the questions below about the reading
1.
What is the difference between a directed line segment and a vector in standard position?
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2.
Find the component form and magnitude of the vector v that has initial point (4, -7) and the
terminal point (-1, 5). Then graph the directed line segment of vector v and the
component form of vector v on the same coordinate plane below.
v = ________________________________
‖𝒗‖ = _______________
3.
Describe two different ways you calculate the magnitude of a vector in the coordinate
plane?
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4.
Describe two different ways you can calculate the direction of a vector in the coordinate
plane?
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5.
Explain why you think changing a directed line segment to component form is useful?
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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. Use a different color for each vector.
Label each vector and write final solution in component
form.
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. Check your solution
with the component form answer on the
left.
Component form
3v = < _____________ >
-1/2w = < _____________ >
B) Find -½ w algebraically. Check your
solution with the component form
answer on the left.
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.
Label each vector and write final solution in component
form.
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. Check
your solution with the component form
answer on the left.
Component form
v + 2w = < _____________ >
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.
Label each vector and write final solution in component
form.
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. Check your
solution with the component form answer on
the left.
Component form
2w -v = < _____________ >
Unit 9 Lesson 1
(Long Block Only)
Unit 9 Lesson 1 Problem Set
Directions: In problems 1-6, use the vectors in the figure to the left to graph the
following vectors. Label all vectors and use graph paper.
1.
v+w
4. w – v
2. u + v
3. 3u
5. 3v + u – 2v
6. 2u – 3v + w
Directions: In problems 7-12, determine if each statement is true or false Explain your
reasoning.
7. a + b = f ______
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10.
b+k+g=a
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13.
8. k + g = f
______
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9. c+ h = k
______
___________________________
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11. a – b = g - k ______
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___________________________
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12. a + b + k + g = 0 ______
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What is the magnitude of your displacement when you follow directions that tell you to
walk 225 m in one direction, make a 90 ̊ turn to the left and walk 350 m, then make a 30 ̊
turn to the right and walk 125 m?
Unit 9 Lesson 1
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