Use vectors and vector addition to solve real-world problems LT 8.7 Use vectors and vector addition to solve real-world problems

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.

Example 1: Identifying Equal and Parallel Vectors Identify each of the following. A. equal vectors B. parallel 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.

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

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 Wind

Step 2: Write the vector for the airplane in component form. < 375.877, 136.808> Step 3: Write the vector for the wind in component form. <0, 60> 70° 400 Airplane 20° y x S W E N 60 Wind

Step 4 Find and sketch the resultant vector Step 4 Find and sketch the resultant vector . Add the components of the airplane’s vector and the wind vector. <375.877, 196.808> 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). 424.3 at N 62° E

Example 6 An Olympic marathon runner starts running down a street at a bearing of N 34ºW for 12 miles then turns down a street heading due south for 14.2 miles where she crosses the finish line. In what direction should a helicopter leave the runner’s starting point in order to get to the finish line? How far will the helicopter fly?