lesson 6 menu 1.determine whether the dilation is an enlargement, reduction, or congruence...
TRANSCRIPT
1. Determine whether the dilation is an enlargement, reduction, or congruence transformation for a scale factor of r= 2/3.
2. Determine whether the dilation is an enlargement,reduction, or congruence transformation for a scale factor of r = 24.
3. Determine whether the dilation is an enlargement,reduction, or congruence transformation for a scale factor of r = 1.
4. Find the measure of the dilation image of AB with the scale factor r = – 2 and AB = 3.
5. Find the measure of the dilation image of AB with the scale factor r = 5/7 and AB = 3/5.
• vector
• magnitude
• direction
• standard position
• component form
• Find magnitudes and directions of vectors.
• Perform translations with vectors.
• equal vectors
• parallel vectors
• resultant
• scalar
• scalar multiplication
Write Vectors in Component Form
Find the change of x values and the corresponding change in y values.
Component form of vector
Simplify.
Magnitude and Direction of a Vector
Find the magnitude and direction of for S(–3, –2) and T(4, –7).
Find the magnitude.
Distance Formula
Simplify.
Use a calculator.
Magnitude and Direction of a Vector
Graph to determine how to find the direction. Draw a right triangle that has as its hypotenuse and an acute angle at S.
Magnitude and Direction of a Vector
Simplify.
Substitution
Use a calculator.
tan S
Magnitude and Direction of a Vector
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°.
1. A
2. B
3. C
4. D
A. 4; 45°
B. 5.7; 45°
C. 5.7; 225°
D. 8; 135°
Find the magnitude and direction of for A(2, 5) and B(–2, 1).
Translations with Vectors
Answer: First graph quadrilateral HJLK.
Connect the vertices for
quadrilateral H'J'L'K'.
Next translate each vertex by , 5 units right and 5 units down.
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 .
1. A
2. B
3. C
4. D
A. B.
C. D.
Graph the image of triangle ABC with vertices A(7, 6), B(6, 2), and C(2, 3) under the translation of
Add Vectors
Graph the image of ΔEFG with vertices E(1, –3), F(3, –1), and G(4, –4) under the translationand .Graph ΔEFG.
Method 1 Translate two times.
Translate each vertex 4 units left and 2 units up.
Then translate each vertex of 2 units right and 3 units up. Label the image ΔE'F'G'.
Translate ΔEFG by a. Thentranslate this image of ΔEFG by b.
Answer: Notice that the vertices for the image are the same for either method.
Add Vectors
Method 2 Find the resultant, and then translate.
Translate each vertex 2 units left and 5 units up.
Add a and b.
A. A
B. B
C. C
D. D
Graph the image of ΔABC with vertices A(0, 6), B(–1, 2), and C(–5, 3) under the translation by and
A. B.
C. D. none of the
above
A. 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.
Solve Problems Using Vectors
Solve Problems Using Vectors
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.
The resultant direction of the canoe is about 36.9° south of due east.
Answer: Therefore, the resultant vector is 5 miles per hour at 36.9° south of due east.
Solve Problems Using Vectors
B. 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?
Solve Problems Using Vectors
Use scalar multiplication to find the magnitude of the vector for the river.
Magnitude of
Simplify.
Solve Problems Using Vectors
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.
Solve Problems Using Vectors
A. A
B. B
C. C
D. D
A. Direction is about 60.3° south of due east with a velocity of about 8.1 miles per hour.
B. Direction is about 60.3° south of due east with a velocity of about 11 miles per hour.
C. Direction is about 29.7° south of due east with a velocity of about 8.1 miles per hour.
D. Direction is about 29.7° south of due east with a velocity of about 11 miles per hour.
KAYAKING Suppose a person is kayaking due east across a lake at 7 miles per hour. If the lake is flowing south at 4 miles an hour, what is the resultant direction and velocity of the canoe?