Vector Addition and Subtraction

Question

Suppose you walk 18.0 m straight west and then 25.0 m straight north. (If you represent the two legs of the walk as vector displacements A and B, as in the figure below, then this problem asks you to find their sum R = A + B.)

Three vectors labeled A, B and R are on an x y coordinate plane. Vector A begins from the origin and extends to the left along the negative x-axis. The tail of vector B begins at the head of vector A and extends straight up in the positive y-direction. The tail of vector R is located at the origin and the head of vector R is located at the same position as the head of vector B. Next to the coordinate system, the equation A + B = R and a coordinate system depicting North, East, South, and West with an arrow facing upward toward the north are also shown.

How far, in meters, are you from your starting point? (Enter a number.)

___ m

What is the compass direction of a line connecting your starting point to your final position measured in degrees west of north? (Enter a number.)

  ___°

Answer 1

This becomes a trigonometry problem. If you have a right triangle with two sides known, then the third side can be found with the Pythagorean relationship. (a^2 + b^2 = c^2). In this case c, the hypotenuse, is the length of vector R. Thus, the magnitude is sqrt( 18^2 + 25^2 ) = 30.8 m.

To find the angle, you want to think back to SOH-CAH-TOA (acronym for Sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, and tan θ = opposite / adjacent). Here tangent would be the best choice because it doesn't depend on your answer to part 1. Therefore, tan θ = 25 / 18 . This implies that θ = tan^-1( 25/18) = 45.3 degrees.