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Lesson Menu Five-Minute Check (over Lesson 8–6) CCSS Then/Now New Vocabulary Example 1:Represent Vectors Geometrically Key Concept: Vector Addition Example 2:Find the Resultant of Two Vectors Example 3: Write a Vector in Component Form Example 4: Find the Magnitude and Direction of a Vector Key Concept: Vector Operations Example 5: Operations with Vectors Example 6:Real-World Example: Vector Applications

Over Lesson 8–6 5-Minute Check 1 A.50.1 B.44.6 C.39.3 D.35.9 Find s if the measures of ΔRST are m  R = 63, m  S = 38, and r = 52.

Over Lesson 8–6 5-Minute Check 2 A.21.3 B.24.1 C.29 D.58 Find m  R if the measures of ΔRST are m  S = 122, s = 10.8, and r = 5.2.

Over Lesson 8–6 5-Minute Check 3 A.12.7 B.10.8 C.9.62 D.8.77 Use the measures of ΔABC to find c to the nearest tenth.

Over Lesson 8–6 5-Minute Check 4 A.21° B.19° C.18° D.16° Use the measures of ΔABC to find m  B to the nearest degree.

Over Lesson 8–6 5-Minute Check 5 A.21 mi B.18 mi C.16 mi D.15.5 mi 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?

CCSS Content Standards G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Mathematical Practices 1 Make sense of problems and persevere in solving them. 4 Model with mathematics.

Then/Now You used trigonometry to find side lengths and angle measures of right triangles. Perform vector operations geometrically. Perform vector operations on the coordinate plane.

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

Example 1 Represent Vectors Geometrically A. Use a ruler and a protractor to draw each vector. Include a scale on each diagram. = 80 meters at 24° west of north Using a scale of 1 cm : 50 m, draw and label an 80 ÷ 50 or 1.6-centimeter arrow 24º west of the north- south line on the north side. Answer:

Example 1 Represent Vectors Geometrically B. Use a ruler and a protractor to draw each vector. Include a scale on each diagram. = 16 yards per second at 165° to the horizontal Using a scale of 1 cm : 8 yd/s, draw and label a 16 ÷ 8 or 2-centimeter arrow at a 165º angle to the horizontal. Answer:

Example 1 Using a ruler and a protractor, draw a vector to represent feet per second 25  east of north. Include a scale on your diagram. A. B. C. D.

Concept

Example 2 Find the Resultant of Two Vectors Subtracting a vector is equivalent to adding its opposite. a b Copy the vectors. Then find

Example 2 Method 1Use the parallelogram method. Find the Resultant of Two Vectors –b a a Step 1, and translate it so that its tail touches the tail of.

Example 2 Step 2Complete the parallelogram. Then draw the diagonal. Find the Resultant of Two Vectors a – b –b a

Example 2 Method 2Use the triangle method. Find the Resultant of Two Vectors –b a Step 1, and translate it so that its tail touches the tail of.

Example 2 Find the Resultant of Two Vectors Step 2Draw the resultant vector from the tail of to the tip of –. Answer: a – b a –b a – b

Example 2 Copy the vectors. Then find A.B. C.D. a – b ba

Example 3 Write a Vector in Component Form Write the component form of.

Example 3 Find the change of x-values and the corresponding change in y-values. Component form of vector Simplify. Write a Vector in Component Form

Example 3 Write the component form of. A. B. C. D.

Example 4 Find the Magnitude and Direction of a Vector Step 1Use the Distance Formula to find the vector’s magnitude. Simplify. Use a calculator. Find the magnitude and direction ofDistance Formula (x 1, y 1 ) = (0, 0) and (x 2, y 2 ) = (7, –5)

Example 4 Find the Magnitude and Direction of a Vector Graph, its horizontal component, and its vertical component. Then use the inverse tangent function to find θ. Step 2Use trigonometry to find the vector’s direction.

Example 4 Find the Magnitude and Direction of a Vector Definition of inverse tangent Use a calculator.The direction of is the measure of the angle that it makes with the positive x-axis, which is about 360 – 35.5 or So, the magnitude of is about 8.6 units and the direction is at an angle of about 324.5º to the horizontal. Answer:

Example 4 A.4; 45° B.5.7; 45° C.5.7; 225° D.8; 135° Find the magnitude and direction of

Concept

Example 5 Operations with Vectors Solve Algebraically Find each of the following for and. Check your answers graphically. A. Check Graphically

Example 5 Operations with Vectors Solve Algebraically Find each of the following for and. Check your answers graphically. B. Check Graphically

Example 5 Operations with Vectors Solve Algebraically Find each of the following for and. Check your answers graphically. C. Check Graphically

Example 5 A. B. C. D.

Example 6 Vector Applications 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 per hour, what is the resultant speed and direction of the canoe? Draw a diagram. Let represent the resultant vector.

Example 6 Vector Applications The component form of the vector representing the velocity of the canoe is  4, 0 , and the component form of the vector representing the velocity of the river is  0, –3 . The resultant vector is  4, 0  +  0, –3  or  4, –3 , which represents the resultant velocity of the canoe. Its magnitude represents the resultant speed.

Example 6 Use the Distance Formula to find the resultant speed. Distance Formula (x 1, y 1 ) = (0, 0) and (x 2, y 2 ) = (4, –3) The resultant speed of the canoe is 5 miles per hour. Vector Applications

Example 6 Use trigonometry to find the resultant direction. Use a calculator. Vector Applications Definition of inverse tangent The resultant direction of the canoe is about 36.9° south of due east. Answer: Therefore, the resultant speed of the canoe is 5 mile per hour at an angle of about 90° – 36.9° or 53.1° east of south.

Example 6 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 speed of the canoe?

End of the Lesson