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The Pythagorean Theorem is a fundamental concept in geometry. It relates the lengths of the sides of a right triangle and is crucial in calculating the magnitude of vectors. When you have a vector in the form of its components, like the ones given in the exercise, you can think of it as forming a right triangle with its x-component and y-component representing the two legs of the triangle. The magnitude of the vector is the hypotenuse of this triangle.

Mathematically, the Pythagorean Theorem states:

• For a right triangle with legs of lengths \( a \) and \( b \), the hypotenuse \( c \) is given by \( c = \sqrt{a^2 + b^2} \).

In vector terms, we use the same idea: The magnitude of a vector with components \( a \) and \( b \) is \( \sqrt{a^2 + b^2} \). This formula is used repeatedly in the solutions provided to find the lengths of the vectors from their components.

Vectors in a coordinate system often have two primary pieces called components. These components describe how far the vector stretches along the x-axis and the y-axis. In vector notation, \( \vec{A} = a\hat{x} + b\hat{y} \), the term \( a\hat{x} \) is the x-component and \( b\hat{y} \) is the y-component. Understanding this is vital for vector addition, subtraction, and other operations.

Components:

• The x-component shows the horizontal distance or movement.
• The y-component indicates the vertical distance or movement.

When solving for the magnitude of a vector, knowing its components allows us to apply the Pythagorean Theorem effectively. If one component is zero, like \( \vec{A} = -2\hat{y} \), it makes the calculations straightforward since the vector lies entirely along one axis.

A coordinate system is essential to precisely define vectors and their components. It is like a map that provides a frame where we can accurately track an object's location and movement. In two-dimensional space, the Cartesian coordinate system is most commonly used and consists of the x-axis and y-axis, intersecting at the origin (0,0).

Benefits of Using a Coordinate System:

• Allows precise descriptions of positions and vectors.
• Helps in visually representing mathematical problems and solutions.
• Makes it easier to apply formulas, like the calculation of vector magnitudes.

By representing vectors in terms of their components within a coordinate system, it makes mathematical operations, such as finding magnitudes or directions, manageable and systematic. Understanding the coordinate system is critical for following the process in the given exercise where vectors are expressed as combinations of their components.