If $straight theta$ is the angle between $straight A with rightwards arrow on top space and space straight B with rightwards arrow on top$ , then show that $open vertical bar straight A with hat on top minus straight B with hat on top close vertical bar space equals space 2 sin straight theta over 2.$

Solution

On evaluating $open vertical bar straight A with hat on top minus straight B with hat on top close vertical bar$
Therefore,
$open vertical bar straight A with hat on top minus straight B with hat on top close vertical bar space equals space square root of open vertical bar straight A with hat on top close vertical bar squared plus open vertical bar straight B with hat on top close vertical bar squared minus 2 open vertical bar straight A with hat on top close vertical bar open vertical bar straight B with hat on top close vertical bar cosθ. end root$
$equals square root of 1 plus 1 minus 2 space cosθ end root$
$equals square root of 2 left parenthesis 1 minus cosθ right parenthesis end root equals space square root of 2.2 sin squared straight theta over 2 end root$
$equals 2 sin straight theta over 2$