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Word Problems: Vectors - Right Triangle Relationships
In order to solve problems involving angles of elevation and depression, it is necessary to
 
 
A typical problem involving vectors and right triangles gives us information about a vector and/or its components. We are asked to find information about unmeasured components and angles. Note that we will represent vectors either by using bold type v or by using the vector notation .
 
Suppose a plane is taking off at a rate of 200 mph with an angle of 35° to the ground. What is the rate at which it is receding from the ground? What is its ground speed?
 
First we make a diagram. In this case, the plane’s path makes a 35° angle with the ground. We call this vector p.
 
 
The hypotenuse of the triangle is given a length of 200 mph which is the magnitude of the vector p describing the plane’s path. The vector representing climbing vertically away from the ground is c and the vector representing the ground speed is g. The right angle is formed between vectors c and g which are components of p.
 
 
An easy way to identify which vectors are the components and which vector is the resultant (or sum of those two components) is to trace their "head-to-tail" relationship.
 
When adding vectors, we start by placing the head of the first vector (g) at the original of a co-ordinate system.
 
At its tail we place the origin of a second co-ordinate system and the head of the second vector (c).
 
The resultant (p) is the vector that begins at the head of vector 1 (g) and ends at the tail of vector 2 (c) as shown in the diagram to the right.
 
In the diagram shown below the known magnitude (||p|| = 200 mph) and angle (q = 35º) are labeled as well as the unknown vectors corresponding to the situation: the magnitude of the climbing speed of the plane by c, and the magnitude of the ground speed of the plane by g.
 
 
Using the right triangle ratios sine and cosine we have:
 
 
Thus the climbing and ground speeds are approximately 114.7 mph and 163.8 mph respectively and are the vertical and horizontal components of the resultant vector p.

Let's Practice

Question #1
AudioA force of 100 pounds is required to pull a 400 pound block up a ramp at a constant speed. What angle does the ramp make with respect to the ground?


Question #2
Audio
A plane is traveling at 400 mph bearing 20° (bearings are measured from North in a clockwise manner). How fast is the plane traveling North? How fast is the plane traveling East?
 



Try These
Question #1
AudioAn airplane is flying North at 500 mph when a wind of 20 mph comes up out of the East. What is the new speed and direction of the plane if no correction is made for the wind?


Question #2
AudioAn object is pulled up a ramp at an angle of 17° with the ground level. If the force required to move this object up the incline at a constant speed is 200 pounds, how much does the object weigh?



As you can see, this type of problem requires a diagram of a carefully labeled right triangle. The measures of the magnitudes of any given vectors and their components as well as any known angles should be labeled. Trigonometric ratios often help us set up an equation which can then be solved for the requested missing information.. If the two legs of the triangle are a part of the problem, it is a tangent ratio. If the hypotenuse is part of the problem, it is either a sine or cosine ratio.

M Ransom

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