maliaseth0

2022-01-24

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

caoireoilns

Expert

Let $\stackrel{\to }{A}$ be the position vector of A and $\stackrel{\to }{B}$ be the position vector of B. A position vector is a vector that points from the origin to a particular point.
If you plot $\stackrel{\to }{A},\stackrel{\to }{B}$ and $\stackrel{\to }{AB}$ , then you can easily notice that $\stackrel{\to }{B}=\stackrel{\to }{A}+\stackrel{\to }{AB}$ using the triangle rule of addition of vectors.
In this question, $\stackrel{\to }{A}=4\stackrel{\to }{i}+2\stackrel{\to }{j}-6\stackrel{\to }{k}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\stackrel{\to }{B}=9\stackrel{\to }{i}-\stackrel{\to }{j}+3\stackrel{\to }{k}$. So
$9\stackrel{\to }{i}-\stackrel{\to }{j}+3\stackrel{\to }{k}=\left(4\stackrel{\to }{i}+2\stackrel{\to }{j}-6\stackrel{\to }{k}\right)+\stackrel{\to }{AB}$, , thus meaning that
$\stackrel{\to }{AB}=\left(9-4\right)\stackrel{\to }{i}+\left(-1-2\right)\stackrel{\to }{j}+\left(3+6\right)\stackrel{\to }{k}=5\stackrel{\to }{i}-3\stackrel{\to }{j}+9\stackrel{\to }{k}$.

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