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

Over Lesson 8–6

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 50.1

B. 44.6

C. 39.3

D. 35.9

Find s if the measures of ΔRST are mR = 63, mS = 38, and r = 52.

Over Lesson 8–6

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 21.3

B. 24.1

C. 29

D. 58

Find mR if the measures of ΔRST are mS = 122, s = 10.8, and r = 5.2.

Over Lesson 8–6

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

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

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 21°

B. 19°

C. 18°

D. 16°

Use the measures of ΔABC to find mB to the nearest degree.

Over Lesson 8–6

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

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?

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.

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.

• vector

• standard position

• component form

• magnitude

• direction

• resultant

• parallelogram method

• triangle method

Write Vectors in Component Form

Write the component form of .

Write Vectors in Component Form

Find the change of x-values and the corresponding change in y-values.

Component form of vector

Simplify.

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

Write the component form of .

A.

B.

C.

D.

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 thevector’s magnitude.

Distance Formula

Simplify.

Use a calculator.

Find the 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.

Step 2 Use trigonometry to find the vector’sdirection.

Find the Magnitude and Direction of a Vector

Simplify.

Substitution

Use a calculator.

tan S

Definition of inverse tangent

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°.

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

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).

Vector Addition and Subtraction

Subtracting a vector is equivalent to adding its opposite.

Vector Addition and Subtraction

Method 1 Use the parallelogram method.

Step 2 Complete the parallelogram.

Vector Addition and Subtraction

Step 3 Draw the diagonal of the parallelogram from the initial point.

Vector Addition and Subtraction

Method 2 Use the triangle method.

Vector Addition and Subtraction

Answer:

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. B.

C. D.

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.

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.

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.

A. A

B. B

C. C

D. D

A B C D

0% 0%0%0%

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?