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. F x = F cos A

Working with Vectors Mathematically For the y-axis, this is the sine since the y- axis is the opposite side of the triangle. F Y = 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 x- component of the force is; F x = F cos A = 30 N x cos 20 o F x = 30 N x = 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 y- component of the force is; F Y = F sin A = 30 N x sin 20 o F x = 30 N x = 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