AI Tutor
If a rescue drone flies east at 12 meters per second while a northern crosswind blows it at 5 meters per second, where does it actually end up?
To find its true speed and path, we must combine these forces into a single resultant vector and find its size and heading.
How do we combine separate vector arrows into one overall displacement?

Combining Components

A vector is a quantity that has both a magnitude (length) and a direction. To add multiple vectors algebraically, you simply add their corresponding horizontal (xx) and vertical (yy) components.
Given two vectors uβƒ—=⟨ux,uy⟩\vec{u} = \langle u_x, u_y \rangle and vβƒ—=⟨vx,vy⟩\vec{v} = \langle v_x, v_y \rangle, their sum Rβƒ—=uβƒ—+vβƒ—\vec{R} = \vec{u} + \vec{v} has components Rβƒ—=⟨ux+vx,uy+vy⟩\vec{R} = \langle u_x + v_x, u_y + v_y \rangle.
Now that we have the combined horizontal and vertical steps RxR_x and RyR_y, how do we find the direct straight-line distance?

Calculating Magnitude

The magnitude of a vector, written as βˆ₯Rβƒ—βˆ₯\|\vec{R}\|, is its geometric length. Because the xx and yy axes meet at a 90∘90^\circ angle, the vector forms the hypotenuse of a right triangle with legs ∣Rx∣|R_x| and ∣Ry∣|R_y|.
Using the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), the magnitude formula is:
βˆ₯Rβƒ—βˆ₯=Rx2+Ry2\|\vec{R}\| = \sqrt{R_x^2 + R_y^2}