Pythagorean theorem
Summary
In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. In physics it gives the magnitude of a resultant from perpendicular vector components, R = √(Rx² + Ry²). It holds for right angles, so resolve other vectors into perpendicular components first.
Detail
Meaning: In a right triangle, the square of the hypotenuse (the side opposite the 90° angle) equals the sum of the squares of the other two sides. In physics it gives the magnitude of a vector from its perpendicular components, or the resultant of two perpendicular vectors.
Equation: c² = a² + b², where c is the hypotenuse and a and b are the legs (m). For vectors, R = √(Rx² + Ry²), where R is the resultant magnitude and Rx and Ry are perpendicular components, in the quantity's own unit (m, m/s or N).
On the MCAT: Expect the speed of a projectile from vx and vy, net displacement after perpendicular legs of a walk, or net force from two perpendicular forces. Common triples (3-4-5, 5-12-13) allow quick arithmetic without a calculator. The direction then comes from tan θ = opposite/adjacent.
Common confusion: Adding the components (3 + 4 = 7) instead of squaring, summing and taking the root (5). The resultant of perpendicular vectors is smaller than the sum of their magnitudes, and it is always longer than either component alone. The theorem holds for right triangles; for vectors at other angles, resolve them into perpendicular components first or use the law of cosines.
Sources
- Physics
- Fundamentals of Physics
Written for the MCAT and checked before it was published. For study reference only. Spotted an error? Let us know.