Scalars are quantities that only have a magnitude, like the radius in a circle, or the distance between two cities. Resultant and Components Resultant X Component Component Vectors are quantities that need to be defined by both a magnitude and a direction. Here is a visual representation between a scalar and a vector: Scalar Vector 5 Meters Long 5 Meters East StickMan Physics 640 The scalar only says that something is 5 meters long, while the vector says that it is 5 meters long and in the easterly direction. A vector will be given with a direction based on one of the vertical directions (North or South), and then an angle and one of the horizontal directions (East or West) to tell you which way to angle it. The first step of solving any vector problem is to make a sketch. North West East South A vector will be given with a direction based on one of the vertical directions (North or South), and then an angle and one of the horizontal directions (East or West) to tell you which way to angle it. The first step of solving any vector problem is to make a sketch. For example, a plane was traveling N5817'E.Start at due North and angle your vector 5817' to the East. North 5817' West East South The arrow end of a vector is referred to as the tip, and the other end as the tail. Vectors are often not drawn in the coordinate plane, but they can be, and you will occasionally encounter problems that give you coordinates for the vectors, and you will have to use your other trigonometric skills to determine angles and other information. The notation in this lesson is not necessarily typical of navigation or aviation. We use these examples so you can picture what it means to find the resultant of adding two vectors. It's possible you'll get other kinds of bearings or headings as well--occasionally a problem will be talking about ascension and descent instead of NESW, or will be giving bearings from a totally different reference point. Unfortunately, there are a plethora of ways to show the direction of a vector, so one needs to be flexible when thinking about it. Vector Addition Most problems you'll be asked will involve more than one vector. The resultant is the vector sum of two vectors. watch this short video: Finding a resultant vector 7 m/s a + b = c 5 + 7 = c c = 5+7 -1 Watch later Share 5 m/s Golden rule for vector triangles: Draw one vector at a time, starting each one at the end of the previous one. Watch on YouTube tano=7 HP 5 0=54 5 m/s 7 m/s b A. If the vectors are parallel to each other, finding the resultant is easy...add the magnitudes. So if both vectors are traveling in the same direction, the result is a much bigger magnitude, B. if the parallel vectors are opposed to each other, then you'll need to take the difference. A. If the vectors are parallel to each other, finding the resultant is easy...add the magnitudes. So if both vectors are traveling in the same direction, the result is a much bigger magnitude, B. if the parallel vectors are opposed to each other, then you'll need to take the difference. Starting from home, I drive 6 miles due East. I stop for a break, then drive West for 2 miles. How far am I from home? The vectors would be parallel to each other, but in opposite directions. The vector East is larger, the answer will still be positive in the East direction. 6-2=4 4 miles East C. When vectors are given at angles to one another, it's a little more complicated. There are two ways to find the resultant, the parallelogram method and the triangle method. Two Vectors 22 Parallelogram Method Triangle Method In the parallelogram method, put the vectors tail to tail and make a parallelogram, solving for the diagonal. In the triangle method, y put the vectors tip to tail, and solve for the third side of the triangle. (Be careful not to combine the two: Putting the vectors tail to tail and solving for the third side of the triangle will result in a very different answer!) The triangle method has the advantage of allowing the use all the tools in the trigonometric toolbox, especially the Law of Sines and the Law of Cosines (which will also help find the direction of the resultant.) Aviation Problems Remember to use reference angles. Pay attention to where angles are being measured from in case you need to convert them for use in formulas. Check to make certain yo sense when you get it. Generally vector problems are practical, so you won't usually come up with impossible results. Ex. 1 A plane is flying west at 200 mph. The wind begins blowing S30W at 25 mph. What is the ground speed and the new direction of the plane's path? After taking into account what effect wind speed has, we calculate ground speed. For example, a plane traveling 100 mph, flying headfirst into a 20 mph wind, has a ground speed of only 80 After taking into account what effect wind speed has, we calculate ground speed. For example, a plane traveling 100 mph, flying headfirst into a 20 mph wind, has a ground speed of only 80 mph. Watch this short video: Fx Navigation Vector Directions - "Heading" or Lrn Fzx Vector Watch later Share radians NAVIGATION Directions Watch on YouTube S1 Make a sketch, and use that to make a triangle: 22 Use the tail to tip method! It doesn't matter which order, but NO tail to tail which results in the wrong resultant. S2 Now let's puzzle out any angles we can. We know that the plane was flying due West, and that the wind was S30W. If we treat the plane's path like the x axis here, we can figure out that S30W would be 120 from that. So we'll fill that in. 200 S2 Now let's puzzle out any angles we can. We know that the plane was flying due West, and that the wind was S30W. If we treat the plane's path like the x axis here, we can figure out that S30W would be 120 from that. So we'll fill that in. 25 120 200 S3 Solve the triangle, using the Law of Cosines and the Law of Sines. Once you solve the triangle, you then need to convert the angle of the resultant into a number in the fashion we've been using in this lesson (S?W). In real aviation, a heading is usually measured from due North in a clockwise direction, so that a plane heading due West would be heading 270. Some convention in physics problems measures vectors clockwise from the East instead. Whenever you are presented with a vector problem make sure that you know the measuring convention being used! In the problems for this lesson stick with the North, East, West, South convention we've been using from above. Go ahead and solve this problem, it will be one of the problems you need to submit below, but save all your work, I'll be asking you questions about the other steps as well. Additional Resources: 10-minute Video: Bearings vs Direction - Trigonometry Word Problems worection JG Bearings vs Direction Trigonometry Word Pityrection Watch later Share 20 W N N30E 120 E W Watch on YouTube Video: Bearing Problems & Navigation S Grading for this lesson: 30 E Vector NAVIGATION Directions Please show your work. The teacher will look at your work and give you advice on any that you miss. You are not required to submit your drawings for each scenario, but submitting them will allow for better teacher feedback. If you would rather write your work on paper, instead of typing, upload a file for the teacher to see, click here. A plane is flying west at 200 mph. The wind begins blowing S30W at 25 mph. 1. What is the ground speed of the plane now? (nearest tenth) 2. When you solve the triangle, what is the smallest angle? (nearest hundredth) 3. What is the largest angle of the triangle? 4. What is the remaining angle? (nearest hundredth) 5. What is the direction of the plane's new path? (The answer will be written using compass points such as N17W.)