Procedure
1. Draw a diagram, adding the components graphically.
2. Choose x and y axes. Choose them in a way, if possible, that will make your work easier. (For example, choose one axis along the direction of one of the vectors so that a vector will have only one component.)
3. Resolve each vector into its x and y components, showing each component along its appropriate (x or y) axis as a (dashed) arrow.
4. Calculate each component (when not given) using sines and cosines. If A1 is the angle vector V1 makes with the x axis then
V1x = V1cosA1
V1y = V1sinA1
Pay careful attention to signs: any component that points along the negative x or negative y axes gets a negative sign.
5. Add the x components together to obtain the x component of the resultant. Repeat for y:
Vx = V1x + V2x + V3x + ...
Vy = V1y + V2y + V3y + ...
This is the answer: the components of the resultant vector.
6. If you want to determine the magnitude and direction of the resultant vector, use the following equations:
R = sqrt[(Vx)exp2 + (Vy)exp2]
tan A1 = Vy/Vx
The vector diagram you already drew helps obtain the correct position (quadrant) of A1.
Exercises
An airplane trip involves three legs, with two stopovers. The first leg is due east for 620 km. The second leg is due southeast for 440 km. The third leg is 53 degrees south of west for 550 km.
1. Determine the plane's total displacement by graphical means, using the head-to-tail method of vector addition. Be sure to include a length:magnitude scale and directional indices, as well as both acting and adding diagrams.
2. Determine the plane's total displacement by algebraic/trigonometric means, using the method of vector component resolution and addition. Show all work, including sketches of the vectors (and their respective components) referenced to a Cartesian field.
3. Create a vector addition problem involving at least five concurrent vectors and solve. Allow your creativity to flourish!
