Transcript Document

Lesson 9-6
Vectors
Transparency 9-6
5-Minute Check on Lesson 9-5
Determine whether the dilation is an enlargement, a reduction or a
congruence transformation based on the given scaling factor.
1. r = ⅔
2. r = - 4
R
E
3. r = 1
CT
Find the measure of the dilation image of AB with the given scale
factor
4.
AB = 3, r = - 2
5. AB = 3/5, r = 5/7
6.
Standardized Test Practice: Determine the
6
scale factor of the dilated image
A
1
--3
B
1
- --3
C
3
D
-3
Click the mouse button or press the
Space Bar to display the answers.
3/7
Objectives
• Find magnitudes and directions of vectors
• Perform translations with vectors
Vocabulary
• Vector – a quantity that has both magnitude and
direction
• Vector magnitude – its length (use distance formula
to find magnitude)
• Vector direction – measure of angle the vector forms
with x-axis
• Component form – ordered pair representation of a
vector <∆x, ∆y>
• Resultant – the sum of two vectors
• Scalar – a positive constant
• Scalar multiplication – multiplying a vector by a
scalar
Vectors
y
Magnitude is found by
using the distance
formula
Vector addition: add
corresponding
components
Scalar multiplication:
multiply each
component by the
scalar value.
Negative value
reverses direction
Direction is measured
in reference to the xaxis. It is usually
found using tan-1 .
x° - Direction
x positive x-axis is 0°
positive y-axis is 90°
negative x-axis is 180°
negative y-axis is 270°
In navigation, we use
north as 0 degrees
and go around clockwise.
Most practical aspect for physics and engineering classes – wind velocity,
friction, drag, lift, etc can all be represented by vectors
Write the component form of
Find the change of x values
and the corresponding
change in y values.
Component form of vector
Simplify.
Answer: Because the magnitude and direction of a vector
are not changed by translation, the vector
represents the same vector as
Write the component form of
Answer:
Find the magnitude and direction of
and T(4, –7).
for S(–3, –2)
Find the magnitude.
Distance Formula
Simplify.
Use a calculator.
Graph
to determine how to find the direction.
Draw a right triangle that has
as its hypotenuse
and an acute angle at S.
tan S
Sub
Simplify.
A vector in standard position that is equal to
forms a
–35.5° degree angle with the positive x-axis in the fourth
quadrant. So it forms a
angle with
the positive x-axis.
Answer:
has a magnitude of about 8.6 units and a
direction of about 324.5°.
Find the magnitude and direction of
and B(–2, 1).
Answer:  5.7; 225°
for A(2, 5)
Graph the image of quadrilateral HJLK with vertices
H(–4, 4), J(–2, 4), L(–1, 2) and K(–3, 1) under the
translation of v
First graph quadrilateral HJLK.
Next translate each
vertex by , 5 units right
and 5 units down.
Connect the vertices for
quadrilateral
.
Answer:
Graph the image of triangle ABC with vertices A(7, 6),
B(6, 2), and C(2, 3) under the translation of v
Answer:
CANOEING Suppose a person is canoeing due east
across a river at 4 miles per hour. If the river is flowing
south at 3 miles an hour, what is the resultant direction
and velocity of the canoe?
The initial path of the canoe is due east, so a vector
representing the path lies on the positive x-axis 4 units
long. The river is flowing south, so a vector representing
the river will be parallel to the negative y-axis 3 units long.
The resultant path can be represented by a vector from
the initial point of the vector representing the canoe to the
terminal point of the vector representing the river.
Use the Pythagorean Theorem.
Pythagorean Theorem
Simplify.
Take the square root of each side.
The resultant velocity of the canoe is 5 miles per hour.
Use the tangent ratio to find the direction of the canoe.
Use a calculator.
Answer: Therefore, the resultant vector is 5 miles per hour
at 36.9° south of due east.
CANOEING Suppose a person is canoeing due east
across a river at 4 miles per hour. If the current reduces
to half of its original speed, what is the resultant
direction and velocity of the canoe?
Use scalar multiplication to find the magnitude of the vector
for the river.
Magnitude of
Simplify.
Next, use the Pythagorean Theorem to find the magnitude
of the resultant vector.
Pythagorean Theorem
Simplify.
Take the square root of each side.
Then, use the tangent ratio to find the direction of the canoe.
Use a calculator.
Answer: If the current reduces to half its original speed,
the canoe travels along a path approximately 20.6°
south of due east at about 4.3 miles per hour.
KAYAKING Suppose a person is kayaking due east
across a lake at 7 miles per hour.
a. If the lake is flowing south at 4 miles an hour, what is
the resultant direction and velocity of the canoe?
Answer: Resultant direction is about 29.7° south of due
east; resultant velocity is about 8.1 miles per hour.
b. If the current doubles its original speed, what is the
resultant direction and velocity of the kayak?
Answer: Resultant direction is about 48.8° south of due
east; resultant velocity is about 10.6 miles
per hour.
Summary & Homework
• Summary:
– A vector is a quantity with both magnitude and
direction
– Magnitude is the distance between the two
component vectors (Pythagorean Thrm)
– Vectors can be used to translate figures on the
coordinate plane
• Homework:
– pg 503-504; 15-17, 24-28, 47-50