Displacement vector 𝐀 points due east and has a magnitude of 2.00 km . Displacement vector 𝐁 poi

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Displacement vector 𝐀 points due east and has a magnitude of...

Displacement vector $\overrightarrow{\mathbf{A}}$ points due east and has a magnitude of $2.00 \mathrm{km} .$ Displacement vector $\overrightarrow{\mathbf{B}}$ points due north and has a magnitude of 3.75 km. Displacement vector $\overrightarrow{\mathbf{C}}$ points due west and has a magnitude of $2.50 \mathrm{km} .$ Displacement vector $\overrightarrow{\mathbf{D}}$ points due south and has a magnitude of $3.00 \mathrm{km} .$ Find the magnitude and direction (relative to due west) of the resultant vector $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}+\overrightarrow{\mathbf{D}}$.

Key Concepts

Vector Addition

Vector addition is the process of combining two or more vectors to determine a single resultant vector. This technique involves summing up corresponding components of each vector, and it is fundamental in physics and engineering for understanding cumulative effects like displacement, force, or velocity.

Component Decomposition

Component decomposition involves breaking down a vector into its horizontal (x-axis) and vertical (y-axis) parts. This method simplifies the process of vector addition, as it allows the problem to be handled like any problem involving scalar quantities, using algebraic methods to find the sum of the components.

Resultant Vector

The resultant vector is the final vector obtained after the addition of two or more vectors. It represents the overall effect in terms of magnitude and direction, and it is typically calculated by combining the vector components and using methods like the Pythagorean theorem and trigonometric functions.

Magnitude Calculation

Calculating the magnitude of a vector involves finding the square root of the sum of the squares of its components, as stated by the Pythagorean theorem. This step is crucial in determining the overall size or length of the resultant vector in a coordinate system.

Direction Calculation

Direction calculation involves determining the angle between the resultant vector and a reference axis, often using inverse trigonometric functions like the arctan function. This process allows us to precisely understand the orientation of the vector relative to a defined direction, such as due west in this problem.

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