A force vector 𝐅1 points due east and has a magnitude of 200...

A force vector $\overrightarrow{\mathbf{F}}_{1}$ points due east and has a magnitude of 200 newtons. A second force $\overrightarrow{\mathbf{F}}_{2}$ is added to $\overrightarrow{\mathbf{F}}_{1} .$ The resultant of the two vectors has a magnitude of 400 newtons and points along the east/west line. Find the magnitude and direction of $\overrightarrow{\mathbf{F}}_{2} .$ Note that there are two answers.

Key Concepts

Ambiguity in Vector Solutions

Sometimes, vector equations can yield more than one solution, typically because the conditions provided (such as a net magnitude and direction) can be met by different distributions of the individual vector components. This ambiguity is particularly evident when determining an unknown force that, when added to a known force, results in a specified horizontal or vertical resultant, leading to more than one mathematically acceptable answer.

Resultant Force

The resultant force is the single force that has the same effect as the combined forces acting on an object. Its magnitude and direction are determined by summing the vector components of all the individual forces. In problems where the resultant is specified to lie along a particular line (such as the east–west line), this gives key information about the necessary cancellation of any perpendicular components.

Component Analysis

Breaking vectors into components (often along horizontal and vertical axes) simplifies dealing with forces that are not aligned. This approach allows one to work with scalar quantities along each axis independently, ensuring that the overall vector sum correctly reflects both magnitude and direction.

Vector Addition

This concept involves combining two or more vectors by adding their corresponding components. In problems involving multiple forces, the net effect, or resultant, is found by summing the horizontal and vertical components separately. This method is essential for solving problems where the direction and magnitude of the combined forces must be determined.

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