Mathematical Method for Determining Resultants of Vectors that form right triangles! If two component vectors are at right angles to each other, their resultant vector will form the hypotenuse of a right triangle. A B R

Use the Pythagorean Theorem to Find the Magnitude of the Resultant! Recall Pythagorean Theorem: c 2 = a 2 + b 2 A B R Rewrite, substituting in vector symbols: R 2 = A 2 + B 2

Use the Pythagorean Theorem to Find the Magnitude of the Resultant! Rearranging, we get: R = A 2 + B 2 A B R √ NOTE: This equation only works for any two component vectors that are at right angles to each other. It will not work for more than two component vectors!

Use the Tangent Function to Find the Direction of the Resultant! Recognize that we will always need to find the angle (θ) that is adjacent to the TAIL of the resultant vector in order to find the direction! A B R We use:Tan θ = Opposite Adjacent θ = B A

Use the Tangent Function to Find the Direction of the Resultant! Once we have calculated the angle (θ), we can specify the direction of the vector using the angle and the cardinal compass directions (N,S,E,W)! A B R Solving for the angle, θ = Tan -1 B A θ

Example: A Hiker walks 100 meters due West, then 50 meters due South. Find the Hiker’s Displacement. Next, find magnitude: R = A 2 + B 2 √ First, sketch a diagram (it does NOT have to be to scale) this will help determine the final direction: θ R = (100 m) 2 + (50 m) 2 = 112 m √ Lastly, find Ɵ and specify direction : θ = Tan -1 B = Tan -1 ( 50 m/100 m) = 27° A θ = W 27° S R = 112 m W 27°S