5.9. EXERCISES 71

system in which the third component is altitude and the first and second components aremeasured on a line from West to East and a line from South to North. Find the position ofthis airplane one minute later.

Consider the vector (1,2,1) , is the initial position vector of the airplane. As it moves,the position vector changes. After one minute the airplane has moved in the i direction adistance of 100× 1

60 = 53 kilometer. In the j direction it has moved 1

60 kilometer during thissame time, while it moves 1

60 kilometer in the k direction. Therefore, the new displacementvector for the airplane is

(1,2,1)+(

53,

160

,160

)=

(83,

12160

,12160

)Example 5.8.8 A certain river is one half mile wide with a current flowing at 4 miles perhour from East to West. A man swims directly toward the opposite shore from the Southbank of the river at a speed of 3 miles per hour. How far down the river does he find himselfwhen he has swam across? How far does he end up swimming?

Consider the following picture.

43

You should write these vectors in terms of components. The velocity of the swimmer instill water would be 3j while the velocity of the river would be−4i. Therefore, the velocityof the swimmer is −4i+ 3j. Since the component of velocity in the direction across theriver is 3, it follows the trip takes 1/6 hour or 10 minutes. The speed at which he travels is√

42 +32 = 5 miles per hour and so he travels 5× 16 = 5

6 miles. Now to find the distancedownstream he finds himself, note that if x is this distance, x and 1/2 are two legs of a righttriangle whose hypotenuse equals 5/6 miles. Therefore, by the Pythagorean theorem thedistance downstream is √

(5/6)2− (1/2)2 =23

miles.

5.9 Exercises1. The wind blows from the South at 40 kilometers per hour and an airplane which

travels at 400 kilometers per hour in still air is heading East. Find the actual velocityof the airplane.

2. ↑In the above problem, find the position of the airplane after two hours.

3. ↑In the above problem, if the airplane is to travel due east, in what direction shouldit head in order to achieve this?

4. The wind blows from West to East at a speed of 50 miles per hour and an airplanewhich travels at 300 miles per hour in still air is heading North West. What is thevelocity of the airplane relative to the ground? What is the component of this velocityin the direction North?