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In physics, the components method is essential for solving vector problems, especially those involving displacement. Displacement is a vector quantity that considers both magnitude and direction. The components method involves breaking down a vector into parts along the coordinate axes, usually the x-axis and y-axis. This makes it easier to manage and analyze complex vector quantities. To use this method:

• Identify the directions of the components of the vector. In our example, north and east are positive directions.
• Resolve each individual vector into its horizontal (east-west) and vertical (north-south) components.
• Add up the similar components from different vectors. This means summing up all north-south components together and east-west components separately.

In our example, the north-south displacement was the difference between the northward and southward movements. The east-west displacement was straightforward as the motion was purely westward.

The Pythagorean Theorem is a fundamental mathematical principle used to relate the lengths of the sides of a right triangle. This theorem is regularly used in physics to find the magnitude of a resultant vector when two vectors are perpendicular to each other.Let's consider the north-south and east-west components of the displacement as the two shorter sides of a right triangle:

• North-south component: 1.75 km
• East-west component: 2.20 km

According to the Pythagorean Theorem, the square of the hypotenuse (resultant vector or displacement in this case) is equal to the sum of the squares of these two sides:\[ c^2 = a^2 + b^2 \]By plugging in the values, we get:\[ c = \sqrt{(1.75)^2 + (2.20)^2} = 2.79 \text{ km} \]This result shows the magnitude of the resultant displacement as 2.79 km.

The arctangent function is a trigonometric function used to find an angle when given the opposite and adjacent sides of a right triangle. This is key when determining the direction of a resultant vector in vector displacement problems.The direction or angle of the resultant vector is found by:

• Using the ratio of the east-west component to the north-south component.
• Applying the arctangent function to calculate the angle.

In our example, for determining the direction of the resultant displacement, the angle is calculated as:\[ \theta = \arctan\left(\frac{-2.20}{1.75}\right) \]This yields an angle of approximately \(-51.34^\circ\). Since this angle is negative, it means the direction is west of north, clarifying that the resultant displacement points in this direction.

A resultant vector represents the combination of two or more vectors in magnitude and direction. It's critically important in understanding how different displacement vectors combine to show an overall change in position or motion.In calculating a resultant vector:

• Firstly, sum up the components in each direction using the components method.
• Utilize the Pythagorean Theorem to find the magnitude.
• Use trigonometric functions, like the arctangent function, to identify its direction effectively.

In the given exercise, after resolving and summing the components of displacement, we used the Pythagorean Theorem to find that the magnitude was 2.79 km. The direction was determined to be \(51.34^\circ\) west of north. By understanding the resultant vector, one can easily visualize and determine how each individual displacement or vector contributes to the total movement.