Component Form

Pre-Calculus · Axiom Academy

A vector is nothing more than how far it travels sideways and how far it travels up — its two components. Drop a vector onto a grid and ask: how far right, how far up? Those two answers — the horizontal component a and the vertical component b — completely pin the arrow down. We write them stacked together as . a = run (x-direction) · b = rise (y-direction) the same arrow, built from the unit vectors and Most often a vector is given by where it starts and where it ends. Slide from the initial point P_1 to the terminal point P_2 : the run is how much x changed, and the rise is how much y changed. Terminal minus initial — that's it. x_2 - x_1 — subtract the start's x from the end's x . y_2 - y_1 — subtract the start's y from the end's y . Destination minus origin. Flip them and the arrow points the opposite way. Two arrows with the same are equal, wherever they sit on the grid. The arrow moves 4 units right and 5 units up — independent of where you draw it. 3. Recover Length and Direction The two legs and the arrow form a right triangle, so the components hand you everything else. The magnitude is the hypotenuse — straight from the Pythagorean theorem — and the direction angle comes from the legs' ratio. length = hypotenuse of the leg triangle angle from the positive x-axis (mind the quadrant) A clean triangle: length 5 , tilted about above horizontal. only knows a/b 's ratio, so it can land in the wrong quadrant. Use to get the true angle.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.