Let a =2 i - j - k, b = i +3 j - k and c =2 i + j +3 k. Let v be the vector in the plane of the vectors a and b, such that the length of its…

Let $\displaystyle \vec{a}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}}, \vec{b}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\displaystyle \vec{c}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$. Let $\displaystyle \vec{v}$ be the vector in the plane of the vectors $\displaystyle \vec{a}$ and $\displaystyle \vec{b}$, such that the length of its projection on the vector $\displaystyle \vec{c}$ is $\displaystyle \frac{1}{\sqrt{14}}$. Then $\displaystyle |\vec{v}|$ is equal to