The ratio of maximum and minimum magnitudes of the resultant...

The ratio of maximum and minimum magnitudes of the resultant of two vectors $\vec{a}$ and $\vec{b}$ is $3: 1$. Now $|\vec{a}|$ is equal to
(A) $|\vec{b}|$
(B) $2|\vec{b}|$
(C) $3|\vec{b}|$
(D) $4|\vec{b}|$

Key Concepts

Vector Addition

Vector addition involves combining two or more vectors to form a resultant vector. This is a fundamental operation in physics and mathematics, typically visualized using the tip-to-tail method or the parallelogram rule, which helps in understanding how both magnitude and direction are affected when vectors are combined.

Resultant Vector Magnitude

The magnitude of the resultant vector is determined by both the magnitudes of the individual vectors and the angle between them. In particular cases, such as when vectors are aligned in the same direction or directly opposite, the resultant magnitude reduces to the sum or the absolute difference of the magnitudes. This property is essential for analyzing the extreme cases of resultant magnitudes.

Extreme Values in Vector Addition

The extreme values—maximum and minimum—of the resultant vector’s magnitude occur when the vectors are arranged so that their contributions either add constructively or destructively. When vectors are parallel and pointing in the same direction, the resultant is maximal (sum of magnitudes), and when they are anti-parallel, the resultant is minimal (difference of magnitudes).

Algebraic Manipulation in Vector Problems

Algebraic manipulation plays a key role in solving vector problems where relationships between magnitudes are given. By setting up equations based on geometric conditions (like the ratio of maximum to minimum resultant magnitudes), one can use techniques such as cross-multiplication and variable isolation to deduce unknown vector magnitudes.

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