A man in search of his dog drives first 10 mi northeast, then 12 mi straight south, and finally

step 1 We drive 10.0 miles northeast 12.0 miles south, then 8.0 miles 30 degrees north of west. What is

A man in search of his dog drives first 10 mi northeast,...

A man in search of his dog drives first 10 mi northeast, then $12 \mathrm{mi}$ straight south, and finally $8 \mathrm{mi}$ in a direction $30^{\circ}$ north of west. What are the magnitude and direction of his resultant displacement?

Key Concepts

Coordinate System

A coordinate system, usually the Cartesian system, is used to provide a framework for defining the direction and orientation of vectors. Establishing a coordinate system is critical to ensure consistent representation and manipulation of the vector components in a problem.

Resultant Vector

The resultant vector is the single vector that has the same effect as all the individual vectors combined. Determining the magnitude and direction of the resultant vector involves combining the individual components derived from each displacement.

Trigonometric Functions in Vector Decomposition

Trigonometric functions like sine and cosine are used to determine the horizontal and vertical components of a vector based on its magnitude and direction. This technique is crucial for accurately resolving vectors that are not aligned with the coordinate axes.

Component Analysis

Component analysis involves breaking down vectors into their perpendicular horizontal and vertical elements, typically along a Cartesian coordinate system. This simplifies the process of adding vectors by dealing with one-dimensional quantities instead of directly working with magnitudes and angles.

Vector Addition

Vector addition is the process of combining two or more vectors to determine a single resultant vector. This approach is essential when multiple movements in various directions are involved since it provides a way to compute the overall change in position.

Vector Displacement

This concept involves representing a change in position as both a magnitude and a direction. In problems like this, displacement is a vector quantity, meaning that not only the distance traveled matters but also the direction relative to a specified reference frame.

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