AI Tutor
When an airplane flies through gusty crosswinds, drawing scale arrows with a ruler on paper isn't precise enough to prevent a crash. Navigational computers calculate exact flight paths by turning every diagonal motion into simple horizontal and vertical parts.
How do we take a single diagonal vector and chop it into numbers that are easy to work with?

Resolving Vectors into Components

A component is the effective value of a vector along a single coordinate axis, like its horizontal shadow (xx) or vertical shadow (yy). Breaking a vector into these pieces is called resolving the vector.
Because perpendicular axes form a right triangle, basic trigonometry connects them: cosine relates the adjacent side to the hypotenuse, while sine relates the opposite side. For a vector Aβƒ—\vec{A} at an angle ΞΈ\theta measured counterclockwise from the positive xx-axis, the components are Ax=Acos⁑θA_x = A \cos\theta and Ay=Asin⁑θA_y = A \sin\theta.
Now that we can split any diagonal arrow into two flat lines, what happens when multiple vectors pull in different directions at once?

Adding Along the Axes

In 1687, Isaac Newton showed that perpendicular motions act independently of each other. This means you can add all the horizontal effects together without worrying about the vertical effects at all.
To find the resultant vector (the single total vector that replaces all individual vectors), simply sum all the xx-components to get RxR_x, and sum all the yy-components to get RyR_y:
Rx=Ax+Bx+Cx+…R_x = A_x + B_x + C_x + \dots Ry=Ay+By+Cy+…R_y = A_y + B_y + C_y + \dots
We have total horizontal motion RxR_x and total vertical motion RyR_y β€” but how do we turn those two separate numbers back into one single vector with a magnitude and a direction?