8-7 Vectors

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

• Perform vector operations geometrically.

• Perform vector operations on the coordinate plane.

Definition

A

B

Initial point or tail

Terminal point or tip

A vector can be represented as a “directed” line segment, useful in describing paths.

A vector looks like a ray, but it is NOT!!

A vector has both direction and magnitude (length).

Direction and Length

From the school entrance, I went three blocks north.

The distance (magnitude) is:

Three blocks

The direction is:

North

Direction and Magnitude

The magnitude of AB is the distance between A and B.

The direction of a vector is measured counterclockwise from the horizonal (positive x-axis).

B

A45°

60°

N

S

EWA

B

Are vectors really used?

http://www.nbclearn.com/portal/site/learn/science-of-nfl-football

Drawing Vectors

Draw vector YZ with direction of 45° and length of 10 cm.

1.Draw a horizontal dotted line

2.Use a protractor to draw 45°

3.Use a ruler to draw 10 cm

4.Label the points

45°

Y

Z

10 c

m

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:

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.

Resultant = Vector SumA path or trip that consists of several segments

can be modeled by a sequence of vectors. The endpoint of one vector is the origin of the next vector in the chain. The figure shows a ship’s path from point M to point N that consists of five vectors.

M

S

T

U

VN

Resultant (Vector Sum)What is the shortest path from M to N?

Write the vector sum for the boat’s trip starting with MS

M

S

T

U

VN

For resultants (vector sums), the following is true:

XY + YZ = XZ

X

Y

Z

p. 601

Types of Vectors• Parallel vectors have

the same or opposite direction but not necessarily the same magnitude (length)

• Opposite vectors have the same magnitude but opposite direction.

• Equivalent vectors have the same magnitude and direction.

Find the Resultant of Two Vectors

Subtracting a vector is equivalent to adding its opposite.

a bCopy the vectors. Then find

Method 1 Use the parallelogram method.

–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.

a – b

–b

a

Method 2 Use the triangle method.

–b

a

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

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

Vectors on the Coordinate Plane

Write the component form of .

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

Component form of vector

Simplify.

Write the component form of .

A.

B.

C.

D.

Assignment

Worksheet 3-1C

8-7 Vectors day 2

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

• Perform vector operations geometrically.

• Perform vector operations on the coordinate plane.

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)

Step 2 Use trigonometry to find the vector’s direction.

Definition of inverse tangent

Use a calculator.

Answer:

A. 4; 45°

B. 5.7; 45°

C. 5.7; 225°

D. 8; 135°

Find the magnitude and direction of

p. 603

Scalar – a constant multiplied by a vector

Scalar multiplication – multiplication of a vector by a scalar (dilation)

Solve Algebraically

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

Check Graphically

Solve Algebraically

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

Check Graphically

Solve Algebraically

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

Check Graphically

A.

B.

C.

D.

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.

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.

Use trigonometry to find the resultant direction.

Use a calculator. 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.

8-7 Assignment