The position vectors of the points A and B are respectively 3î - 5ĵ + 2k̂ and î + ĵ - k̂. What is the length of AB?

Concept:

Let \(\rm {\rm{\vec a}} = {\rm{x\;\vec i}} + {\rm{y\;\vec j}} + {\rm{z\;\vec k}}\) then the magnitude of the vector of \(\vec a\) = \(\left| {{\rm{\vec a}}} \right| = {\rm{\;}}\sqrt {{{\rm{x}}^2} + {\rm{\;}}{{\rm{y}}^2} + {{\rm{z}}^2}} \)

Calculation:

Given:

Position vectors of the points A and B are respectively 3î - 5ĵ + 2k̂ and î + ĵ - k̂

\(\rm \vec{AB} = \rm \vec{B}-\rm \vec{A}\)

⇒ \(\rm \vec{AB} =\)  (î + ĵ - k̂) - (3î - 5ĵ + 2k̂)

\(\rm \vec{AB}= -2\rm\hat{i} + 6\rm\hat{j} -3\rm\hat{k} \)

Now, the length of \(\vec {AB} = \rm |\vec {AB}|\)

⇒ \(\rm |\vec {AB}|\) = \(\sqrt{(-2)^2+6^2+(-3)^2} = \sqrt{49}=7\)