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

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