Conventionally, vectors are added and subtracted via the addition and subtraction of their corresponding components.
where #V_1y# and #V_2y# refer to the vertical (y) components of vector 1 and vector 2 respectively. Accordingly, #V_1x# and #V_2x# refer to the horizontal (x) components of vector 1 and vector 2 respectively.
#hati# simply designates that the component(s) in its parentheses are operating vertically, while #hatj# designates that the component(s) in its parenthesis are operating horizontally.
Conversly, if you wished to subtract vectors, simply replace every (+) in the previous equation with a (-):
Note that this implies that the second vector maintains its magnitude, but is reversed in direction. This should make sense, since the minus would simply be distributed through both components of the second vector. Thus it is crucial to keep track of the signs of the vector components when adding or subtracting them.
What is a Vector Component?
Vector components are the horizontal and vertical distance coordinates that a vector possesses. This means that they describe the coordinates of the vector's tip, and subsequently the angle between the vector and the horizontal axis.
The velocity vector of a plane and the velocity vector of the wind is given below:
What is the final velocity vector of the plane?
To answer this question, we must add the plane's initial vector to that of the wind's. This will give us the plane's final velocity vector. After all, the wind will push the plane, thus making the plane's final velocity vector a combination of the two.
Thus all we must do is add the y components and the x components respectively.
To convert this to polar form, we simply find the magnitude and direction of the vector.