(a) Two workers are trying to move a heavy crate. One pushes...

(a) Two workers are trying to move a heavy crate. One pushes on the crate with a force $\overrightarrow{\mathbf{A}},$ which has a magnitude of 445 newtons and is directed due west. The other pushes with a force $\overrightarrow{\mathbf{B}}$, which has a magnitude of 325 newtons and is directed due north. What are the magnitude and direction of the resultant force $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ applied to the crate? (b) Suppose that the second worker applies a force $-\overrightarrow{\mathbf{B}}$ instead of $\overrightarrow{\mathbf{B}}$. What then are the magnitude and direction of the resultant force $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$ applied to the crate? In both cases express the direction relative to due west.

Key Concepts

Trigonometric Functions

Trigonometric functions, particularly the tangent and its inverse (arctan), are used to determine the direction of a resultant vector relative to a reference axis. When the components of a vector are known, the angle that the vector makes with one of the axes can be calculated, providing a complete description of its orientation in the plane.

Vector Addition

Vector addition involves combining two or more vectors to determine a resultant vector. This requires accounting for both the magnitudes and directions of the original vectors, and is typically done graphically using the head-to-tail method or analytically by adding corresponding components. This technique is foundational in many fields of physics and engineering for analyzing forces, velocities, and other directional quantities.

Vector Subtraction

Vector subtraction is a process of determining the difference between two vectors by effectively adding a vector with the opposite direction. It allows one to compute relative motions or forces by reversing one vector and then applying vector addition techniques. Understanding vector subtraction is crucial for resolving problems where the net effect of opposing forces is analyzed.

Component Decomposition

Decomposing a vector into its components involves breaking down the vector into perpendicular directions, typically along the coordinate axes. This process simplifies calculations by transforming a vector with a given magnitude and direction into its horizontal and vertical parts. It forms the basis for algebraically adding or subtracting vectors in a two- or three-dimensional space.

Pythagorean Theorem

The Pythagorean theorem is used in vector analysis to compute the magnitude of a resultant vector when the components are perpendicular. When vectors are decomposed into orthogonal components, the magnitude of the resultant vector is determined by taking the square root of the sum of the squares of these components. This method is essential in converting component forms back into an overall magnitude.

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