A woman starts walking from home and walks 6 miles at 40^∘ north of east, then 2 miles at 15^∘ e

step 1 First, let's have u, and this is going to equal 6 cosine 40 degrees, 6 sine 40 degrees, and u eq

A woman starts walking from home and walks 6 miles at 40^∘...

A woman starts walking from home and walks 6 miles at $40^{\circ}$ north of east, then 2 miles at $15^{\circ}$ east of south, then 5 miles at $30^{\circ}$ south of west. If she walked straight home, how far would she have to walk, and in what direction?

Key Concepts

Directional Conventions

Understanding directional conventions, such as 'north of east' or 'east of south', is important for correctly translating angles into standard coordinate directions. This understanding ensures proper assignment of positive or negative values to the components, which is critical when performing vector addition in a two-dimensional plane.

Resultant Displacement

The resultant displacement is the single vector that has the same effect as a series of individual displacements. After summing the individual components of each vector, the magnitude of the resultant vector can be determined using the Pythagorean theorem, and its direction can be calculated using inverse trigonometric functions, typically the arctan function.

Vector Addition

Vector addition is a fundamental concept in physics and mathematics where multiple displacement vectors are combined to obtain a single resultant vector. This principle allows one to determine the cumulative effect of several individual movements, even when they occur in different directions. The overall displacement is found by adding each vector’s components along a common coordinate system.

Components of Vectors

Breaking vectors into their horizontal and vertical components is essential for simplifying calculations. By resolving a vector into its components using sine and cosine functions corresponding to its angle, one can treat movement along perpendicular axes independently, which facilitates the process of vector addition.

Trigonometric Resolution

Trigonometric resolution involves using the sine and cosine functions to compute the magnitude of a vector's projections along the horizontal (x-axis) and vertical (y-axis) directions. This process is crucial for converting directional information given in angular terms into component form, enabling accurate analysis and combination of different vectors.

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