Holt Geometry

8-6 Vectors

Warm UpFind AB.

1. A(0, 15), B(17, 0)

2. A(–4, 2), B(4, –2)

Solve each equation. Round to the nearest tenth or nearest degree.

3. 4.

5. 6.

6.3 3.1

38° 50°

Holt Geometry

8-6 Vectors

Find the magnitude and direction of a vector.

Use vectors and vector addition to solve real world problems.

Objectives

Holt Geometry

8-6 Vectors

vectorcomponent formmagnitudedirectionequal vectorsparallel vectorsresultant vector

Vocabulary

Holt Geometry

8-6 Vectors

The speed and direction an object moves can be represented by a vector. A vector is a quantity that has both length and direction.

You can think of a vector as a directed line segment. The vector below may be named

Holt Geometry

8-6 Vectors

A vector can also be named using component form.The component form <x, y> of a vector lists thehorizontal and vertical change from the initial point to the terminal point. The component form of is <2, 3>.

Holt Geometry

8-6 Vectors

Example 1A: Writing Vectors in Component Form

Write the vector in component form.

The horizontal change from H to G is –3 units.

The vertical change from H to G is 5 units.

So the component form of is <–3, 5>.

Holt Geometry

8-6 Vectors

Example 1B: Writing Vectors in Component Form

with M(–8, 1) and N(2, –7)

Write the vector in component form.

Subtract the coordinates of the initial point from the coordinates of the terminal point.

= <x2 – x1, y2 – y1>

= <2 – (–8), –7 – 1>

= <10, –8>

Substitute the coordinates of the given points.

Simplify.

Holt Geometry

8-6 Vectors

Check It Out! Example 1a

Write the vector in component form.

The horizontal change is –3 units.

The vertical change is –4 units.

So the component form of is <–3, –4>.

Holt Geometry

8-6 Vectors

Check It Out! Example 1b

Write each vector in component form.

the vector with initial point L(–1, 1) and terminal point M(6, 2)

Subtract the coordinates of the initial point from the coordinates of the terminal point.

Substitute the coordinates of the given points.

Simplify.

= <x2 – x1, y2 – y1>

= <6 – (–1), 2 – 1>

= <7, 1>

Holt Geometry

8-6 Vectors

When a vector is used to represent speed in a given direction, the magnitude of the vector equals the speed. For example, if a vector represents the course a kayaker paddles, the magnitude of the vector is the kayaker’s speed.

The magnitude of a vector is its length. The magnitude of a vector is written

Holt Geometry

8-6 Vectors

Example 2: Finding the Magnitude of a Vector

Draw the vector <–1, 5> on a coordinate plane. Find its magnitude to the nearest tenth.

Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. Then (–1, 5) is the terminal point.

Step 2 Find the magnitude. Use the Distance Formula.

Holt Geometry

8-6 Vectors

Check It Out! Example 2

Draw the vector <–3, 1> on a coordinate plane.Find its magnitude to the nearest tenth.

Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. Then (–3, 1) is the terminal point.

Step 2 Find the magnitude. Use the Distance Formula.

Holt Geometry

8-6 Vectors

The direction of a vector is the angle that it makes with a horizontal line. This angle is measured counterclockwise from the positive x-axis. The direction of is 60°.

The direction of a vector can also be given as a bearing relative to the compass directions north, south, east, and west. has a bearing of N 30° E.

Holt Geometry

8-6 Vectors

Example 3: Finding the Direction of a Vector

The force exerted by a skier is given by the vector <1, 4>. Draw the vector on a coordinate plane. Find the direction of the vector to the nearest degree.

Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point.

So mA = tan-1(4) 76°.

Step 2 Find the direction.

Draw right triangle ABC as shown. A is the angle formed by the vector and the x-axis, and .

A C

B

Holt Geometry

8-6 VectorsCheck It Out! Example 3

The force exerted by a tugboat is given by the vector <7, 3>. Draw the vector on a coordinate plane. Find the direction of the vector to the nearest degree.

Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point.

A

B

CStep 2 Find the direction.

Draw right triangle ABC as shown. A is the angle formed by the vector and the x-axis, and

Holt Geometry

8-6 Vectors

Two vectors are equal vectors if they have the samemagnitude and the same direction. For example, . Equal vectors do not have to have the same initial point and terminal point.

Holt Geometry

8-6 Vectors

Two vectors are parallel vectors if they have the same direction or if they have opposite directions. They may have different magnitudes. For example, Equal vectors are always parallel vectors.

Holt Geometry

8-6 Vectors

Example 4: Identifying Equal and Parallel Vectors

Identify each of the following.

A. equal vectors

B. parallel vectors

Identify vectors with the same magnitude and direction.

Identify vectors with the same or opposite directions.

Holt Geometry

8-6 Vectors

Check It Out! Example 4

Identify each of the following.

a. equal vectors

b. parallel vectors

Identify vectors with the same magnitude and direction.

Identify vectors with the same or opposite directions.

Holt Geometry

8-6 Vectors

The resultant vector is the vector that represents the sum of two given vectors. To add two vectors geometrically, you can use the head-to-tail method or the parallelogram method.

Holt Geometry

8-6 Vectors

Holt Geometry

8-6 Vectors

To add vectors numerically, add their components. If = <x1, y1> and = <x2, y2>, then

= <x1 + x2, y1 + y2>.

Holt Geometry

8-6 Vectors

Example 5: Aviation Application

An airplane is flying at a constant speed of 400 mi/h at a bearing of N 70º E. A 60 mi/h wind is blowing due north. What are the plane’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree.Step 1 Sketch vectors for the airplane and the wind.

70°400

Airplane

20°y

x

S

W E

N

60

S

W E

N Wind

Holt Geometry

8-6 Vectors

Example 5 Continued

Step 2 Write the vector for the airplane in component form. The airplane’s vector has a magnitude of 400 mi/h and makes an angle of 20º with the x-axis.

so x = 400 cos20° 375.9.

The airplane’s vector is <375.9, 136.8>.

so y = 400 sin20° 136.8.

Holt Geometry

8-6 Vectors

Step 3 Write the vector for the wind in component form. Since the wind moves 60 mi/h in the direction of the y-axis, it has a horizontal component of 0 and a vertical component of 60. So its vector is <0, 60>.

The resultant vector in component form is <375.9, 196.8>.

Step 4 Find and sketch the resultant vector . Add the components of the airplane’s vector and the wind vector.

<375.9, 136.8> + <0, 60> = <375.9, 196.8>.

Example 5 Continued

Holt Geometry

8-6 Vectors

Step 5 Find the magnitude and direction of the resultant vector. The magnitude of the resultant vector is the airplane’s actual speed (or ground speed).

The angle measure formed by the resultant vector gives the airplane’s actual direction.

Example 5 Continued

+

Holt Geometry

8-6 Vectors

Check It Out! Example 5

What if…? Suppose the kayaker in Example 5 instead paddles at 4 mi/h at a bearing of N 20° E. What are the kayak’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree.

Step 1 Sketch vectors for the kayaker and the current.

20°

70°

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8-6 Vectors

Check It Out! Example 5 Continued

Step 2 Write the vector for the kayaker in component form. The kayaker’s vector has a magnitude of 4 mi/h and makes an angle of 20° with the x-axis.

The kayaker’s vector is <1.4, 3.8>.

so x = 4cos70° 1.4.

so y = 4 sin70° 3.8.

Holt Geometry

8-6 Vectors

Step 3 Write the vector for the current in component form. Since the current moves 1 mi/h in the direction of the x-axis, it has a horizontal component of 1 and a vertical component of 0. So its vector is <1, 0>.

Check It Out! Example 5 Continued

Step 4 Find and sketch the resultant vector . Add the components of the kayaker’s vector and the current’s vector. <1.4, 3.8> + <1, 0> = <2.4, 3.8> The resultant vector in component form is <2.4, 3.8>.

Holt Geometry

8-6 Vectors

Step 5 Find the magnitude and direction of the resultant vector. The magnitude of the resultant vector is the kayak’s actual speed.

The angle measure formed by the resultant vector gives the kayak’s actual direction.

Check It Out! Example 5 Continued

or N 32° E