# What is the magnitude of vector AB if A= (4,2,-6) and B=(9,-1,3)?

Let $\setminus \vec{A}$ be the position vector of A and $\setminus \vec{B}$ be the position vector of B. A position vector is a vector that points from the origin to a particular point.
If you plot $\setminus \vec{A}$, $\setminus \vec{B}$ and $\setminus \vec{A B}$, then you can easily notice that $\setminus \vec{B} = \setminus \vec{A} + \setminus \vec{A B}$ using the triangle rule of addition of vectors. (Try it for two-dimensional vectors!)
In this question, $\setminus \vec{A} = 4 \setminus \hat{i} + 2 \setminus \hat{j} - 6 \setminus \hat{k}$ and $\setminus \vec{B} = 9 \setminus \hat{i} - \setminus \hat{j} + 3 \setminus \hat{k}$. So $9 \setminus \hat{i} - \setminus \hat{j} + 3 \setminus \hat{k} = \left(4 \setminus \hat{i} + 2 \setminus \hat{j} - 6 \setminus \hat{k}\right) + \setminus \vec{A B}$, thus meaning that $\setminus \vec{A B} = \left(9 - 4\right) \setminus \hat{i} + \left(- 1 - 2\right) \setminus \hat{j} + \left(3 + 6\right) \setminus \hat{k} = 5 \setminus \hat{i} - 3 \setminus \hat{j} + 9 \setminus \hat{k}$.