Finding Component Form

Pre-Calculus · Axiom Academy

EXAMPLE Finding Vector Component Form Build the component form of a vector by reading its horizontal and vertical change off the points. A vector runs from the initial point P_1 = (2, 3) to the terminal point P_2 = (5, 7) . Find its component form , then find its magnitude . The horizontal change (3) and vertical change (4) are the legs of a right triangle — and they are exactly the components of . Nice work. You found a vector's component form straight from its endpoints, then measured its length. Component form: subtract initial from terminal — . It's a change, not a position: means "3 right, 4 up," no matter where the vector starts. Magnitude: the length is , so here . The picture: the components are the legs of a right triangle and the vector is its hypotenuse — that's why the Pythagorean theorem gives the length. Once a vector is in component form, every operation — adding, scaling, finding direction — becomes simple arithmetic on a and b .

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