Lesson 9-6

Vectors

5-Minute Check on Lesson 9-55-Minute Check on Lesson 9-5 Transparency 9-6

Click the mouse button or press the Click the mouse button or press the Space Bar to display the answers.Space Bar to display the answers.

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 3. r = 1

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. Determine the scale factor of the dilated image

Standardized Test Practice:

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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

Vectorsy

xx° - Direction

M

agni

tude

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 x-axis. It is usually found using tan-1 .

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 clock-wise.

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.

Answer: Because the magnitude and direction of a vector are not changed by translation, the vector represents the same vector as

Component form of vector

Simplify.

Write the component form of

Answer:

Find the magnitude and direction of for S(–3, –2) and T(4, –7).

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.

Simplify.

Sub

tan S

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 for A(2, 5) and B(–2, 1).

Answer: 5.7; 225°

Answer:First graph quadrilateral HJLK.

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

Connect the vertices for

quadrilateral .

Next translate each vertex by , 5 units right and 5 units down.

Answer:

Graph the image of triangle ABC with vertices A(7, 6), B(6, 2), and C(2, 3) under the translation of v

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.

Simplify.

Pythagorean Theorem

Take the square root of each side.

Use a calculator.

Then, use the tangent ratio to find the direction of the canoe.

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?

b. If the current doubles its original speed, what is the resultant direction and velocity of the kayak?

Answer: Resultant direction is about 29.7° south of due east; resultant velocity is about 8.1 miles per hour.

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