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Five-Minute Check (over Lesson 8–6) NGSSS Then/Now New Vocabulary Example 1: Write Vectors in Component Form Example 2: Find the Magnitude and Direction of a Vector Key Concept: Equal, Opposite, and Parallel Vectors Key Concept: Vector Addition Example 3: Vector Addition and Subtraction Example 4: Real-World Example: Algebraic Vectors Lesson Menu

Find s if the measures of ΔRST are mR = 63, mS = 38, and r = 52. B. 44.6 C. 39.3 D. 35.9 A B C D 5-Minute Check 1

Find mR if the measures of ΔRST are mS = 122, s = 10.8, and r = 5.2. B. 24.1 C. 29 D. 58 A B C D 5-Minute Check 2

A B C D Use the measures of ΔABC to find c to the nearest tenth. 5-Minute Check 3

A B C D Use the measures of ΔABC to find mB to the nearest degree. 5-Minute Check 4

On her delivery route, Gina drives 15 miles west, then makes a 68° turn and drives southeast 14 miles. When she stops, approximately how far from her starting point is she? A. 21 mi B. 18 mi C. 16 mi D. 15.5 mi A B C D 5-Minute Check 5

MA.912.D.9.3 Use vectors to model and solve application problems. LA.1112.1.6.1 The student will use new vocabulary that is introduced and taught directly. MA.912.D.9.3 Use vectors to model and solve application problems. NGSSS

Find magnitudes and directions of vectors. You used trigonometry to find side lengths and angle measures of right triangles. (Lesson 8–4) Find magnitudes and directions of vectors. Add and subtract vectors. Then/Now

vector standard position component form magnitude direction resultant parallelogram method triangle method Vocabulary

Write the component form of . Write Vectors in Component Form Write the component form of . Example 1

Find the change of x-values and the corresponding change in y-values. Write Vectors in Component Form Find the change of x-values and the corresponding change in y-values. Component form of vector Simplify. Example 1

Write the component form of . A. B. C. D. A B C D Example 1

Find the magnitude and direction of for S(–3, –2) and T(4, –7). Find the Magnitude and Direction of a Vector Find the magnitude and direction of for S(–3, –2) and T(4, –7). Step 1 Use the Distance Formula to find the vector’s magnitude. Distance Formula Simplify. Use a calculator. Example 2

Step 2 Use trigonometry to find the vector’s direction. Find the Magnitude and Direction of a Vector Step 2 Use trigonometry to find the vector’s direction. Graph to determine how to find the direction. Draw a right triangle that has as its hypotenuse and an acute angle at S. Example 2

Definition of inverse tangent Find the Magnitude and Direction of a Vector tan S Substitution Simplify. Definition of inverse tangent Use a calculator. Example 2

Find the 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°. Example 2

A B C D Find the magnitude and direction of for A(2, 5) and B(–2, 1). Example 2

Concept

Concept

Subtracting a vector is equivalent to adding its opposite. Vector Addition and Subtraction Subtracting a vector is equivalent to adding its opposite. Example 3

Method 1 Use the parallelogram method. Vector Addition and Subtraction Method 1 Use the parallelogram method. Step 2 Complete the parallelogram. Example 3

Step 3 Draw the diagonal of the parallelogram from the initial point. Vector Addition and Subtraction Step 3 Draw the diagonal of the parallelogram from the initial point. Example 3

Method 2 Use the triangle method. Vector Addition and Subtraction Method 2 Use the triangle method. Example 3

Vector Addition and Subtraction Answer: Example 3

A. B. C. D. A B C D Example 3

Algebraic Vectors 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. Example 4

Use the Pythagorean Theorem. Pythagorean Theorem Algebraic 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. Example 4

The resultant direction of the canoe is about 36.9° south of due east. Algebraic Vectors 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. Example 4

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? 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. A B C D Example 4

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