Vectors as Component Practice and Application - Solutions
In using ground speed, we mean the resulting speed of the plane as observed form the
ground after the effect of the wind. For example, if a plane travels 620 mph into a head
wind of 20 mph, its ground speed is 600 mph.
1. Given a vector w with magnitude 12 and direction 142°, give its component form.
2. An airplane has an airspeed of 520 mph and a bearing of 115°. The wind blows from
the east at 37 mph. Find the plane’s ground speed and bearing.
3. A pilot wants his plane to head N 20°W at 500 mph (ground speed). There is a wind
blowing S 10° E at 45 mph. What direction should he head and at what airspeed? Draw a
picture of this situation and set up a vector equation before you solve this problem.
4. A fishing boat leaves port and travels at a constant rate of 15 knots, heading N 20 E
for 6 hours. It then changes course to N 30 W for 4 hours. Finally it changes course to
N 10 W for 10 hours. At this point, the fishermen have a complete catch and wish to
return directly to port. What direction should they head and how long does it take them?
We find displacement vectors:
1. Traveling for 6 hours, they go 90 knots in direction N 20 E
d1  90cos(70 ),90sin(70 )  30.78,84.57
2. Traveling for 4 hours, they go 60 knots in direction N 30 W
d 2  60cos(120 ),60sin(120 )  30,51.96
3. Traveling for 10 hours, they go 150 knots in direction N 10 W
d3  150cos(100 ),150sin(100 )  26.05,147.72
5. A ferry boat leaves the east bank of the Mississippi River with a compass
heading of 300° and is traveling 10 mph relative to the water (speed in still
water). Assuming the river is running directly south at this point at 4 mph,
determine the boats actual course, the time it takes the cross the river if it is ½
mile wide, and how far up or downstream it lands.
6. Determine the magnitude and resulting direction of the answer to the following:
12,  3  2 3,7  6 8,10
7. Vector A makes a 39° angle with the positive x-axis and has a magnitude of 60. It is
added to vector B that makes a 125° angle to the positive x-axis with a magnitude of 27.
a. Determine the magnitude and direction of the sum A  B .
b. Find the magnitude and direction of the vector C such that A  C  B .