Question

Can the magnitude of the resultant of the two vectors be greater than the sum of magnitude of individual vector?

Answer 1

Answer on Question #71180 – Math – Linear Algebra

Question

Can the magnitude of the resultant of the two vectors be greater than the sum of magnitude of individual vector?

Solution

Let $a = |\vec{a}|, b = |\vec{b}|$, then

$$\left| \vec {a} + \vec {b} \right| = \sqrt {\left| \vec {a} \right| ^ {2} + \left| \vec {b} \right| ^ {2} + 2 \left| \vec {a} \right| \left| \vec {b} \right| \cos \theta} = \sqrt {a ^ {2} + b ^ {2} + 2 a b \cos \theta},$$

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

The maximum value of $|\vec{a} + \vec{b}|$ is reached when $\cos \theta = 1$:

$$\left| \vec {a} + \vec {b} \right| _ {m a x} = \sqrt {a ^ {2} + b ^ {2} + 2 a b} = a + b,$$

which is actually the sum of magnitudes of individual vectors.

Answer: no.

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