You used trigonometry to find side lengths and angle measures of right triangles.

• Perform vector operations geometrically.

• Perform vector operations on the coordinate plane.

• vector

• magnitude

• direction

• resultant

• parallelogram method

• triangle method

• standard position

• component form

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:

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:

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:

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:

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.

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.

Find the Resultant of Two Vectors

Subtracting a vector is equivalent to adding its opposite.

a b

Copy the vectors. Then find

Method 1 Use the parallelogram method.

Find the Resultant of Two Vectors

–b

a

–b

a

Step 1 , and translate it so that its tail touches the tail of .

Step 2 Complete the parallelogram. Then draw the diagonal.

Find the Resultant of Two Vectors

a – b

–b

a

Method 2 Use the triangle method.

Find the Resultant of Two Vectors

–b

a

Step 1 , and translate it so that its tail touches the tail of .

Find the Resultant of Two Vectors

Step 2 Draw the resultant vector from the tail of to the tip of – .

Answer: a

–b

a – b

Find the Resultant of Two Vectors

Step 2 Draw the resultant vector from the tail of to the tip of – .

Answer:

a – b

a

–b

a – b

Copy the vectors. Then find

A. B.

C. D.

a – ba – b

a – b a – b

ba

Copy the vectors. Then find

A. B.

C. D.

a – ba – b

a – b a – b

ba

Write a Vector in Component Form

Write the component form of .

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

Component form of vector

Simplify.

Write a Vector in Component Form

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

Component form of vector

Simplify.

Write a Vector in Component Form

Write the component form of .

A.

B.

C.

D.

Write the component form of .

A.

B.

C.

D.

Find the Magnitude and Direction of a Vector

Step 1 Use the Distance Formula to find thevector’s magnitude.

Simplify.

Use a calculator.

Find the magnitude and direction of

Distance Formula

(x1, y1) = (0, 0) and (x2, y2) = (7, –5)

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 2 Use trigonometry to find the vector’sdirection.

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 324.5. So, the magnitude of is about 8.6 units and the direction is at an angle of about 324.5º to the horizontal.

Answer:

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 324.5. So, the magnitude of is about 8.6 units and the direction is at an angle of about 324.5º to the horizontal.

Answer:

A. 4; 45°

B. 5.7; 45°

C. 5.7; 225°

D. 8; 135°

Find the magnitude and direction of

A. 4; 45°

B. 5.7; 45°

C. 5.7; 225°

D. 8; 135°

Find the magnitude and direction of

Operations with Vectors

Solve Algebraically

Find each of the following for and . Check your answers graphically.A.

Check Graphically

Operations with Vectors

Solve Algebraically

Find each of the following for and . Check your answers graphically.B.

Check Graphically

Operations with Vectors

Solve Algebraically

Find each of the following for and . Check your answers graphically.C.

Check Graphically

A.

B.

C.

D.

A.

B.

C.

D.

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.

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.

Use the Distance Formula to find the resultant speed.

Distance Formula

(x1, y1) = (0, 0) and (x2, y2) = (4, –3)

The resultant speed of the canoe is 5 miles per hour.

Vector Applications

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:

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.

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?

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?