Vector Addition How to resolve vectors into components

Vector Addition How to resolve vectors into components and algebraically add them using a spreadsheet.

Vector Addition Perpendicular Vectors: A resultant (R) can be gotten by placing two vectors (V 1 and V 2) head to tail and solving for the resultant using Pythagorean Theorem. Y R X V 1 Qxr V 2 The direction is an angle counterclockwise from the positve X axis: Qxr

Equations Pythagorean Theorem: Inverse Tangent IF V 2 is negative (q. XR is in quadrant 2 or 3), add 180 to reverse the direction.

What about when vectors are not perpendicular to each other? Y X A B Vectors should still be placed head to tail and a resultant drawn from the tail of the first to the head of the last.

Adding Non-Perpendicular Vectors Y A X Resultant B Although graphically this technique works it is difficult to apply the math necessary to determine a resultant, particularly if there are more than two vectors that must be added.

Vector Decomposition • Therefore we will use the method of vector decomposition to add vectors. – This is an approach that uses the components of the vectors to perform vector addition rather than the vectors themselves. Vector components of a vector V are the two vectors parallel to the X and Y coordinate axes whose resultant is equal to the original vector V.

How do we calculate components? For a given vector V at angle qxr Y X V Vy qx Vx Vy = Y component of V Vx = X component of V qx = angle measured counterclockwise from the + X axis.

Adding Non-Perpendicular Vectors Y VXA X A VYA B VXB Now after determining the components of A and B. VYB

Adding Non-Perpendicular Vectors Y VXA X VYA VYB VXB And replacing the original vectors with them.

Adding Non-Perpendicular Vectors Y X Resultant VYA VYB VXB The components are summed head to tail. VXA

Combining the individual vectors to get the components of the Resultant: RX and RY VXB + VXA = RX VYB + VYA = RY RY R = Resultant RX

Equations Pythagorean Theorem: Inverse Tangent IF RX is negative (q. XR is in quadrant 2 or 3), add 180 to reverse the direction.

You will now use a spreadsheet to…. • Calculate the components of any vector given its magnitude and direction • Calculate the magnitude of the resultant of any number of vectors • Calculate the final direction of the resultant given the sum of the components of each vector.

Tips • An IF function is used to calculate the direction of the resultant. – A cell containing an IF function can have two possible values. – It uses a logical test to check a given value using a greater than or less than comparison. • IF the test is true the result of the function is calculated using the “value if true” part of the function • IF the test is false the result of the function is calculated using the “value if false” part of the function