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The Pythagorean Theorem is a fundamental principle in geometry. It relates the lengths of the sides of a right triangle. A right triangle has one angle measuring exactly 90 degrees.
The theorem states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Mathematically, it is expressed as:

• \[ c^2 = a^2 + b^2 \]

In this formula, \(c\) represents the hypotenuse, while \(a\) and \(b\) represent the other two sides.
When applied to vectors at right angles to each other, this theorem can help in finding the resultant vector's magnitude or one of the vectors' magnitudes when the other magnitudes are known.

A right triangle is a special type of triangle characterized by having one angle that is 90 degrees. This property makes it especially useful in vector mathematics when solving real-world problems.
Since the sides of a right triangle are related by the Pythagorean Theorem, it allows for quick calculations and deductions.

• The side opposite the right angle is known as the hypotenuse, usually the longest side in the triangle.
• The other two sides are called legs and are often involved in calculations using trigonometric functions such as sine, cosine, and tangent.

Right triangles frequently appear in physics and engineering, especially when decomposing vectors which are at right angles, similar to the vectors \( \vec{A} \) and \( \vec{B} \) in this exercise.
Recognizing a right triangle allows us to apply these geometrical principles easier, making it simpler to solve for unknown values.

The concept of vector magnitude refers to the length or size of a vector. A vector is a quantity that has both direction and magnitude.
Calculating the magnitude involves determining the overall size of the vector, regardless of its direction.For a vector \( \vec{C} = \langle x, y \rangle \), its magnitude is given by the formula:

• \[ |\vec{C}| = \sqrt{x^2 + y^2} \]

In scenarios where vectors are at right angles, calculating their combined magnitude uses a similar approach as demonstrated with the Pythagorean Theorem.
This relates directly to understanding how two perpendicular vectors, like \( \vec{A} \) and \( \vec{B} \), can combine to form a new vector whose magnitude is determined using this principle.
Knowing how to calculate the magnitude is essential in vector math for simplifying complex physics problems, especially those involving forces or motion.