# If $\rm \vec a$ = 2î + ĵ + k̂ and $\rm \vec b$ = î + 2ĵ + k̂, then the magnitude of their resultant is:

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If $\rm \vec a$ = 2î + ĵ + k̂ and $\rm \vec b$ = î + 2ĵ + k̂, then the magnitude of their resultant is:
1. 2√5
2. 2√6
3. √22
4. None of these.

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Correct Answer - Option 3 : √22

Concept:

The magnitude of a vector $\rm \vec A$ = a1î + a2ĵ + a3k̂ is given as $\rm \left|\vec A\right|=\sqrt{a_1^2+a_2^2+a_3^2}$.

The magnitude of the sum of vectors $\rm \vec A$ and $\rm \vec B$ can also be calculated as $\rm \left|\vec A+\vec B\right|=\sqrt{{\vec A}^2+{\vec B}^2+2\vec A\cdot\vec B}$.

The resultant of a set of vectors acting at a point is simply the algebraic sum of the vectors.

Calculation:

The resultant of the vectors $\rm \vec a$ = 2î + ĵ + k̂ and ​$\rm \vec b$​ = î + 2ĵ + k̂ is:

$\rm \vec r=\vec a + \vec b$ = (2î + ĵ + k̂) + (î + 2ĵ + k̂) = 3î + 3ĵ + 2k̂

Now, $\rm \left|\vec a+\vec b\right|=\sqrt{3^2+3^2+2^2}=\sqrt{22}$.