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Resolution and Composition of Vectors

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.

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Resolution and Composition of Vectors

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  1. Resolution and Composition of Vectors

  2. 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.

  3. 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.

  4. Working with Vectors Mathematically • You are in fact calculating the sides of a 90o (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

  5. Working with Vectors Mathematically • For the y-axis, this is the sine since the y-axis 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.

  6. Working with Vectors Mathematically • Let’s assume that our vector F is 30 N of force. It is acting at an angle of 20o to the x-axis. Based on this information, the x-component of the force is; Fx = F cos A = 30 N x cos 20o Fx = 30 N x 0.940 = 28.2 N

  7. Working with Vectors Mathematically • Let’s assume that our vector F is 30 N of force. It is acting at an angle of 20o to the x-axis. Based on this information, the y-component of the force is; FY = F sin A = 30 N x sin 20o Fx = 30 N x 0.342 = 10.3 N

  8. 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.

  9. Working with Vectors Mathematically • To add or compose two vectors into one resultant, you will use Pythagorean’s theorem.

  10. 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

  11. 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 = 36.9o

  12. 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 36.9o N of E

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