Vectors

Warm Up – Find the reference angle and the value of the given angle Warm Up – Find the reference angle and the value of the given angle. What other angle between 0 and 360 would give the same value?

Objectives Find the magnitude and direction of a vector. Use vectors and vector addition to solve real world problems.

Vocabulary vector component form magnitude direction equal vectors parallel vectors resultant vector

Indicate each path by an arrow. From home, walk 1 mi E, 1 mi E, 1 mi E, and 1 mi E. 2. From home, walk 1 mi E, 1 mi N, 1 mi W, and 1 mi S. 3. From home, walk 1 mi E, 1 mi E, 1 mi E, and 1 mi N. 4. From home, walk 1 mi E, 1 mi E, 1 mi N, and 1 mi N.

How far are you away from home? 1mi 2. 1mi 1. 4 mi 0 mi 1mi 4. 1mi 3.

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 (tail) (head) The speed and direction an object moves can be represented by a vector.

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

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>.

Example 1B: Writing Vectors in Component Form Write the vector in component form. with M(–8, 1) and N(2, –7) = <10, –8>

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>.

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

Resultant Vectors 1mi 2. 1mi 1. 1mi 4. 1mi 3.

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

1mi 1.

1mi 2.

1mi 3.

1mi 4.

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

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 distance or pythagorean thm.

1mi 1.

1mi 2.

1mi 3.

1mi 4.

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 (or E 60° N).

Name the angle measure in compass direction. W 223°

Name the angle measure in compass directions. W 300°

Find the Direction of a Vector B C

Finding magnitude and direction of a vector. Given

1mi 1.

1mi 2.

1mi 3. 18.43º

1mi 4. 45º

1. A person walks 200 meters at 27º North of East. Example: Draw the indicated vector and show the components into which it is resolved. 1. A person walks 200 meters at 27º North of East. N E S W 27º 200m Y X

Example: Calculate and draw the resultant of the vectors Example: Calculate and draw the resultant of the vectors. Include direction. A boat sails South at 10m/s (relative to the water) while a current carries it due west at 6.7 m/s. What is its velocity relative to an observer fixed on the dock? N E S W 6.7m/s 10m/s

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. 70° 400 Airplane 20° S W E N 60 Wind

N E S W Check It Out! Example 5 Suppose the kayaker paddles at 4 mi/h at a bearing of N 20° E and the current is running at a bearing E40ºS at 1 mi/h. What are the kayak’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. N E S W 1 4 70º 40º

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

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.

Lesson Quiz: Part I Round angles to the nearest degree and other values to the nearest tenth. 1. Write with S(–5, 2) and T(8, –4) in component form. 2. Write with magnitude 12 and direction 36° in component form. 3. Find the magnitude and direction of the vector <4, 5>. <13, –6> <9.7, 7.1> 6.4; 51°

Lesson Quiz: Part II 4. Find the sum of the vectors <2, –4> and <3, 6>. Then find the magnitude and direction of the resultant vector. 5. A boat is heading due east at a constant speed of 35 mi/h. There is an 8 mi/h current moving north. What is the boat’s actual speed and direction? <5, 2>; 5.4; 22° 35.9 mi/h; N 77° E