Question

two vector a & b are added.prove that the magnitude of resultant vector cannot be greater than (a+b) and smaller than (a-b)

Answer 1

Answer on Question #43461 – Math – Analytic Geometry

Two vector $a$ & $b$ are added. Prove that the magnitude of resultant vector cannot be greater than $(a + b)$ and smaller than $(a - b)$.

Solution

The magnitude of resultant vector is

$$\boxed{a + b} = \sqrt{a^2 + b^2 + 2ab \cos \theta},$$

where $\theta$ is the angle between the vectors $\vec{a}$ and $\vec{b}$.

When $\theta$ is zero, then resultant vector has the maximum length, equal to $\sqrt{a^2 + b^2 + 2ab} = \sqrt{(a + b)^2} = |a + b|$.

When $\theta$ is 180 degrees, then resultant vector has the minimum length, equal to $\sqrt{a^2 + b^2 - 2ab} = \sqrt{(a - b)^2} = |a - b|$.

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