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A resultant vector is the single vector which has the same effect as the combined effect of all the vectors acting on a point. Imagine you are pushing a box with two distinct forces. The resultant vector would represent the overall force you apply to the box. In mathematics, to find the resultant of two vectors, you can add them together. This combination of vectors is often achieved by using vector addition and is expressed as \( \vec{R} = \vec{P} + \vec{Q} \). Here, \( \vec{R} \) is the resultant vector of \( \vec{P} \) and \( \vec{Q} \).A helpful way of visualizing this is by drawing the vectors from head to tail in a sequence, where the resultant vector is drawn from the tail of the first vector (\( \vec{P} \)) to the head of the last vector (\( \vec{Q} \)). In the scenario provided, when \( \vec{Q} \) is altered, such as doubled or reversed, it affects the magnitude and direction of the resultant vector \( \vec{R} \), showing the dynamic nature of vector operations.

The magnitude of a vector is essentially its size or length. When you picture a vector on a graph or in a physical space, its magnitude is the distance from the starting point to the endpoint. For instance, if \( \vec{P} = (x, y) \), the magnitude of \( \vec{P} \) is calculated using the formula \( \|\vec{P}\| = \sqrt{x^2 + y^2} \). This gives us a scalar value representing how long or how strong the vector is, independent of its direction. In our exercise, new conditions in the problem state changes to the vectors. When the vector \( \vec{Q} \)'s magnitude is doubled, \( 2\vec{Q} \), it implies the vector stretches to double its length. Similarly, when we reverse \( \vec{Q} \), its magnitude stays the same, but the direction flips, impacting the resultant vector. Understanding how to compute and interpret magnitude lets you better grasp the influence of vector changes in practical problems, such as calculating the ratios \( P^2 : Q^2 : R^2 \) accurately.

The vector equation is a mathematical expression that relates vectors using vector addition, subtraction, and multiplication. It enables us to calculate unknown vector quantities from known ones, providing a logical framework to solve vector-related problems.In the given problem, the vector equation is used to understand how the resultant vector \( \vec{R} \) reacts to changes in \( \vec{Q} \). For example, when \( 2\vec{Q} \) leads to \( 2\vec{R} \), the vector equation is \( \vec{P} + 2\vec{Q} = 2\vec{R} \). Rearranging this equation helps to infer that \( \vec{P} = \vec{Q} \). Additionally, when reversing \( \vec{Q} \), the equation becomes \( \vec{P} - \vec{Q} = 2\vec{R} \), solving this leads to \( 3\vec{P} = 2\vec{Q} \). These equations not only illustrate vector algebra but also uncover relationships between vector magnitudes, eventually assisting in figuring out their ratios. Mastering vector equations helps solve more complex vector operations confidently.