Resolution and Composition of Vectors Working with Vectors
- Slides: 12
Resolution and Composition of Vectors
Working with Vectors Mathematically • Given a single vector, you may need to break it down into its x and y components. • This requires mathematical operations including the use of sine and cosine.
Working with Vectors Mathematically • Lets assume that you have a vector F at angle A to the x-axis. You would break it down as shown below.
Working with Vectors Mathematically • You are in fact calculating the sides of a 90 o (or right triangle). • For the x-axis, this is the cosine since the x-axis is the adjacent side of the angle. Fx = F cos A
Working with Vectors Mathematically • For the y-axis, this is the since the yaxis is the opposite side of the triangle. FY = F sin A • You must draw the triangle to know which side is the x component and which is the y component.
Working with Vectors Mathematically • Let’s assume that our vector F is 30 N of force. It is acting at an angle of 20 o to the x-axis. Based on this information, the xcomponent of the force is; Fx = F cos A = 30 N x cos 20 o Fx = 30 N x 0. 940 = 28. 2 N
Working with Vectors Mathematically • Let’s assume that our vector F is 30 N of force. It is acting at an angle of 20 o to the x-axis. Based on this information, the ycomponent of the force is; FY = F sin A = 30 N x sin 20 o Fx = 30 N x 0. 342 = 10. 3 N
Working with Vectors Mathematically • If you have two vectors and need to know how they will work together, you must add the two vectors to find the resultant. • You learned how to do this a last class using graphical means. Now you will learn how to do it mathematically.
Working with Vectors Mathematically • To add or compose two vectors into one resultant, you will use Pythagorean’s theorem.
Working with Vectors Mathematically • To determine the angle of the vector you must use the information you have; namely the opposite and adjacent sides. • The tangent of the angle is found by dividing the opposite side divided by the adjacent side. Tan = opposite side = 3 = 0. 75 adjacent side 4
Working with Vectors Mathematically • However, the tangent of the angle is not what we are looking for. We are looking for the angle itself. To find this we must take the arc tangent (Tangent -1). Angle A = arctan (3/4) = arctan (0. 75) Angle A = 36. 9 o
Working with Vectors Mathematically • To give the final answer, the vector that is composed of the two vectors we were given is; Resultant Vector = 5 at 36. 9 o N of E
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