2.2 Adding Vectors and Components

Notes

Perpendicular vectors combine with the Pythagorean theorem. Any vector can be split into perpendicular components.

• Resultant of perpendicular vectors $A$ and $B$: $\displaystyle R = \sqrt{A^2 + B^2}$, direction $\displaystyle \theta = \tan^{-1}\left(\frac{B}{A}\right)$.
• Components of a vector $V$ at angle $\theta$ above the horizontal: $V_x = V\cos\theta$, $V_y = V\sin\theta$.
• To add any vectors: break each into $x$ and $y$ parts, add the parts, then recombine.
• Example: 3 m east plus 4 m north gives $\displaystyle R = \sqrt{9 + 16} = 5 \text{ m}$ at $\tan^{-1}(4/3) \approx 53^\circ$ north of east.