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Understanding the magnitude of a vector is essential for analyzing motion and forces in physics. The magnitude, often referred to as the length or size of a vector, measures how far the vector 'reaches' in a particular direction. In our example, vector \( \vec{A} \) is given as 3.00 units, and vector \( \vec{B} \) is 4.00 units. To visualize it, imagine you are walking 3 steps east and then 4 steps south; the total distance you covered in each direction corresponds to the magnitudes of \( \vec{A} \) and \( \vec{B} \) respectively.

Even though vectors can be in two or three dimensions, determining their magnitude in a two-dimensional plane involves a straightforward application of the Pythagorean theorem—a method you will encounter in the next section.

The direction of a vector is equally as important as its magnitude. It's the 'where' that complements the 'how much' of the magnitude. Direction can be described using various units, such as degrees, radians, or even by reference to the cardinal points (north, east, south, and west). For instance, in our exercise, \( \vec{A} \) is pointing along the positive x-axis, which is considered to be 0 degrees or pointing due east.

On the other hand, \( \vec{B} \) points along the negative y-axis, which corresponds to 270 degrees—or due south. Knowing both the magnitude and the direction enables us to accurately describe the vector's position and to understand vector addition and subtraction, which lead to resultant vectors with their unique magnitudes and directions.

The Pythagorean theorem is a cornerstone of geometry, critical to understanding vectors in physics. It relates the lengths of the sides of a right-angled triangle. Applied to vector analysis, we often use it to find the resultant magnitude when we add or subtract vectors, as done in our exercise.

The formula is: \[ a^2 + b^2 = c^2 \], where \(a\) and \(b\) are the lengths of the two shorter sides of the triangle, and \(c\) is the length of the longest side, or the hypotenuse. When \( \vec{A} \) and \( \vec{B} \) are added head-to-tail, they form a right-angled triangle with their resultant vector. Calculating this resultant magnitude involves squaring the individual vector lengths, adding them together, and then taking the square root of the sum—directly applying the Pythagorean theorem.

Trigonometry, the branch of mathematics dealing with the properties of triangles, is indispensable in physics for solving problems involving vectors. It provides tools for calculating unknown angles or side lengths in right-angled triangles. This is particularly useful when we need to find the direction of a resultant vector after addition or subtraction.

For our vectors, we use the tangent function, which is the ratio of the opposite side to the adjacent side, to figure out the angle of the direction. For instance, the direction angle \( \theta \) of the resultant vector can be found using the equation \( \tan(\theta) = \frac{opposite}{adjacent} \), and solving for \( \theta \) with an inverse tangent function \( \tan^{-1} \). Trigonometry provides us a clear angle of direction from the reference axis, which, combined with the magnitude, gives a complete description of the vector's position and quantity.