Prove that:

$left parenthesis 3 straight A with rightwards arrow on top plus 2 straight B with rightwards arrow on top right parenthesis times left parenthesis 2 straight A with rightwards arrow on top minus 3 straight B with rightwards arrow on top right parenthesis space equals space 6 straight A squared minus 6 straight B squared minus 5 ABcosθ$ , where $straight theta$ is the angle between $straight A with rightwards arrow on top space and space straight B with rightwards arrow on top.$

Solution

By using distributive and commutative law, the dot product can be evaluated as,
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$equals 6 straight A with rightwards arrow on top times straight A with rightwards arrow on top plus 4 straight B with rightwards arrow on top times straight A with rightwards arrow on top minus 9 straight A with rightwards arrow on top times straight B with rightwards arrow on top minus 6 straight B with rightwards arrow on top times straight B with rightwards arrow on top$

$equals 6 straight A squared plus 4 straight A with rightwards arrow on top times straight B with rightwards arrow on top minus 9 straight A with rightwards arrow on top times straight B with rightwards arrow on top minus 6 straight B squared$
$equals 6 straight A squared minus 5 straight A with rightwards arrow on top times straight B with rightwards arrow on top minus 6 straight B squared$

$equals space 6 straight A squared minus 5 ABcosθ minus 6 straight B squared$
That is,
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