The x vector component of a displacement vector 𝐫 has a magnitude of 125 m and points along the

step 1 In this problem, we're given the x and y components of a position vector, and we're asked to fin

The x vector component of a displacement vector 𝐫 has a...

The $x$ vector component of a displacement vector $\overrightarrow{\mathbf{r}}$ has a magnitude of $125 \mathrm{m}$ and points along the negative $x$ axis. The $y$ vector component has a magnitude of $184 \mathrm{m}$ and points along the negative $y$ axis. Find the magnitude and direction of $\overrightarrow{\mathbf{r}}$. Specify the direction with respect to the negative ic

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Key Concepts

Determining Vector Direction Using Trigonometry

The direction of a vector is usually expressed as an angle relative to a coordinate axis. By applying trigonometric functions such as the inverse tangent, one can calculate this angle from the ratio of the vector components, taking into account the appropriate quadrant based on the signs of the components.

Magnitude Calculation Using the Pythagorean Theorem

The magnitude of a vector, given its orthogonal components, can be found using the Pythagorean theorem. This principle states that the square of the magnitude of the vector is equal to the sum of the squares of its components, which provides a method for computing the vector's overall length.

Displacement Vectors

Displacement vectors represent the change in position of an object and are often described in a coordinate system using both magnitude and direction. They can be broken down into components that lie along the coordinate axes, allowing for straightforward analysis of motion or position in two or three dimensions.

Vector Components

Vector components are the projections of a vector along the coordinate axes. By decomposing a vector into its x, y (and possibly z) components, one can analyze and compute various properties of the vector, such as its overall magnitude and direction, using basic mathematical tools.

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