The Trigonometric Way Adding Vectors Mathematically

Components of Vectors Every vector is the resultant of two perpendicular vectors. The perpendicular vectors are called the components of the vector. To find the components, use the following formulas: x-component = Magnitude cos Θ y-component = Magnitude sin Θ Note: The symbol “ Θ” is the Greek letter theta. It is used to abbreviate “the angle.”

Components of Vectors 5.0 N Θ= 37º x-component y-component

Components of Vectors F y =3.0 N F x =4.0 N 5.0 N Θ= 37º F x = 5.0 cos 37º N,F y = 5.0 sin 37º N F = 5.0 N at 37º

Independence of Vector Quantities Perpendicular vector quantities are independent on each other The vector quantity in the x direction does not effect the quantity in the y direction. Example: the current in the river does not effect the time to cross the river, but does effect where you end up

Vector Resolution Vector Resolution is the combining of the x- and y- components into the resultant vector. Use Pythagorean theorem to determine magnitude a 2 + b 2 = c 2 θ adjacent opposite R

Finding the Angle To find the direction, start with the relation: Θ = tan -1 (y/x) The calculator gives the angle the vector makes with the x-axis, not the “proper” angle. If your angle is: Case 1: Angle does not need adjustment Case 2: Proper angle = 180° - Θ Case 3: Proper angle = Θ + 180° Case 4: Proper angle = 360° - Θ 12 34

Finding the Angle

Practice Vector Resolution x-component = 5.0 m y-component = -8.0 m

Adding Vectors at Any Angle 1. Break the vectors into their x and y components. 2. Add each of the x components. 3. Add each of the y components. 4. Determine the length of the resultant using the Pythagorean theorem. 5. Determine the angle using Θ = tan -1 (y/x). Adjust based on the quadrant.  Remember the correct signs of the x and y components.

Adding Vectors at Any Angle Example: Add 12 N at 50 ° and 26 N at 200°