Steps for finding a missing vector:
- Write the vector equation for the problem (drawing a diagram usually helps a lot!)
- Decompose the given vectors and organize known information in a table.
- Write algebraic equations and solve.
- Find magnitude and angle of the resultant if required.
When writing a vector equation for one of these problems, it'll be in a form like
where the missing vector is one of the vectors on the left-hand side of the equation. Note that if the problem is about returning to the starting point, .
For decomposing given vectors, we'll have to define a coordinate system if the problem hasn't, then use the equations
If the final answer is supposed to be in magnitude-angle notation, we'll need to use these equations to get the magnitude and direction:
Step 1. This problem has the plane flying three separate legs (24.0 km at 34.0° south of east, 46.0 km north, and the unknown). We'll call them , , and in order. The resultant is 32.0 km west. So our vector equation is
Step 2. Define a coordinate system so that north is the +y-direction and east is the +x-direction. Then we can decompose vector into x- and y-components. "South of east" in this coordinate system is counterclockwise from the +x-axis, so