Your friend has slipped and fallen. To help her up, you pull...

Your friend has slipped and fallen. To help her up, you pull with a force $\overrightarrow{\mathbf{F}}$, as the drawing shows. The vertical component of this force is 130 newtons, and the horizontal component is 150 newtons. Find (a) the magnitude of $\overrightarrow{\mathbf{F}}$ and $(\mathrm{b})$ the angle $\theta.$

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Key Concepts

Pythagorean Theorem

The Pythagorean theorem is used to find the magnitude of a vector when its perpendicular components are known. By treating the horizontal and vertical components as the legs of a right triangle, the overall force is the hypotenuse of that triangle, calculated as the square root of the sum of the squares of the components.

Trigonometric Functions

Trigonometric functions, particularly the arctan function, are used to determine the angle a vector makes with a reference axis. Given the ratio of the vertical to horizontal components, the inverse tangent function yields the angle of the force vector relative to the horizontal direction, providing insight into its directional properties.

Vector Components

This concept involves breaking a vector into its horizontal and vertical parts. By resolving a force into perpendicular components, one can analyze its effect in each direction independently. This technique is fundamental in physics and engineering for understanding and calculating the net influence of forces along different axes.

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