The value of $|\vec{a}-\vec{b}|$, if two vectors $\vec{a}$ and $\vec{b}$ are such that $|\vec{a}|=1,|\vec{b}|=4$, and $\vec{a} . \vec{b}=5$ is :

$\sqrt{7}$

to find $|\vec{a}-\vec{b}|$

given $|\vec{a}|=1 \quad|\vec{b}|=4 \quad \vec{a} . \vec{b}=5$

We know that $(\vec{a}-\vec{b}) .(\vec{a}-\vec{b})=|\vec{a}-\vec{b}|^2$

$\Rightarrow |\vec{a}|^2+|\vec{b}|^2-2 \vec{a} . \vec{b}=|\vec{a}-\vec{b}|^2$

substituting the values

$\Rightarrow \left(1^2+4^2-2 \times 5\right)=|\vec{a}-\vec{b}|^2$

$\Rightarrow \sqrt{(16+1-10)}=|\vec{a}-\vec{b}|$

$|\vec{a}-\vec{b}|=\sqrt{7}$