Suppose you first walk 12.0 m in a direction 20^∘ west of north and then 20.0 m in a direction 4

step 1 So here in this problem we are given that a person start moving from this point and it moves 20

Suppose you first walk 12.0 m in a direction 20^∘ west of...

Suppose you first walk 12.0 $\mathrm{m}$ in a direction $20^{\circ}$ west of north and then 20.0 $\mathrm{m}$ in a direction $40.0^{\circ}$ south of west. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A and B , as in Figure 3.56, then this problem finds their sum R = A + B .)

Key Concepts

Displacement Vectors

Displacement vectors represent the change in position of an object, indicating both magnitude and direction relative to a starting point. They are used in problems involving motion to succinctly capture both how far and in which direction an object has moved, regardless of the path taken.

Vector Decomposition

Vector decomposition involves breaking a vector into its horizontal and vertical components using trigonometric functions. This is an essential technique that translates directional descriptions (such as those given in compass bearings) into components that can be easily added using standard algebraic methods.

Vector Addition

Vector addition is the process of combining two or more vectors to find a resultant vector, which represents the cumulative effect of the individual displacements. By adding corresponding components of each vector, one can determine both the overall magnitude and the exact direction from the starting point.

Resultant Magnitude and Direction

After calculating the sum of individual vector components, the magnitude of the resultant vector is determined using the Pythagorean theorem, and its direction is found using inverse trigonometric functions. This process yields a clear measure of how far an object is displaced and the compass direction of the net movement.

Trigonometric Relationships in Vectors

Trigonometric functions such as sine and cosine are used to relate the angles provided by directional instructions to the horizontal and vertical components of vectors. Understanding these relationships is crucial for accurately decomposing vectors and determining the final location and its orientation relative to a reference direction.

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