Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in each figure. Round answers to the nearest tenth.
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Verified step by step guidance

1. Identify the magnitudes and directions of the two forces involved.
2. Use the parallelogram rule: draw the two forces as vectors originating from the same point.
3. Complete the parallelogram by drawing lines parallel to each vector from the tip of the other vector.
4. The diagonal of the parallelogram represents the resultant force.
5. Use the law of cosines to calculate the magnitude of the resultant force: \( R = \sqrt{A^2 + B^2 + 2AB\cos(\theta)} \), where \( A \) and \( B \) are the magnitudes of the two forces and \( \theta \) is the angle between them.

Key Concepts

Parallelogram Rule

The parallelogram rule is a method used to determine the resultant vector when two vectors are acting simultaneously. It states that if two vectors are represented as two adjacent sides of a parallelogram, the diagonal of the parallelogram represents the resultant vector in both magnitude and direction. This rule is particularly useful in physics and engineering for analyzing forces and motion.

Vector Magnitude

The magnitude of a vector is a measure of its length or size, often represented as a non-negative number. In the context of forces, the magnitude indicates the strength of the force being applied. To calculate the magnitude of a resultant vector, one typically uses the Pythagorean theorem when the vectors are perpendicular, or other trigonometric methods when they are not.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. These functions are essential in resolving vector components and calculating the resultant force when vectors are not aligned along the same axis. Understanding how to apply these functions allows for accurate calculations of angles and distances in vector analysis.