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Lesson 9-6

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. r = ⅔ 2. r = - 4 3. r = 1

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Lesson 9-6

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  1. Lesson 9-6 Vectors

  2. 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. • 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 R E CT 6 3/7 Standardized Test Practice: 1 --- 3 1 - --- 3 A B 3 - 3 C D C Click the mouse button or press the Space Bar to display the answers.

  3. Objectives • Find magnitudes and directions of vectors • Perform translations with vectors

  4. 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

  5. y x Vectors 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.  Magnitude  x° - Direction Most practical aspect for physics and engineering classes – wind velocity, friction, drag, lift, etc can all be represented by vectors

  6. Write the component form of Answer: Because the magnitude and direction of a vector are not changed by translation, the vector represents the same vector as Example 6-1a Find the change of x values and the corresponding change in y values. Component form of vector Simplify.

  7. Write the component form of Answer: Example 6-1b

  8. Find the magnitude and direction of for S(–3, –2)and T(4, –7). Example 6-2a Find the magnitude. Distance Formula Simplify. Use a calculator.

  9. 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 Example 6-2a Sub Simplify.

  10. 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°. Example 6-2a

  11. Find the magnitude and direction of for A(2, 5)and B(–2, 1). Example 6-2b Answer: 5.7; 225°

  12. 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 Next translate each vertex by , 5 units right and 5 units down. Connect the vertices for quadrilateral . Example 6-3a First graph quadrilateral HJLK. Answer:

  13. Graph the image of triangle ABC with vertices A(7, 6),B(6, 2), and C(2, 3) under the translation of v Example 6-3b Answer:

  14. Example 6-5a 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.

  15. Example 6-5a 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.

  16. Magnitude of Example 6-5a 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. Simplify.

  17. Example 6-5a 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.

  18. Example 6-5b • 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.

  19. 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

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