Let $\vec a$ and $\vec b$ be two non-zero vectors.
Statement-1: (i) $|\vec a +\vec b|^2 = |\vec a|^2 + |\vec b|^2 + 2 (\vec a.\vec b)$
(ii) $|\vec a−\vec b|^2 =|\vec a|^2 + |\vec b|^2 -2 (\vec a.\vec b)$
Statement-2: The greatest and least values of $|\vec a +\vec b|$ are $|\vec a|+|\vec b|$ and $\left||\vec a|-|\vec b|\right|$ respectively.
Answer & explanation
Correct answer: option 2
Clearly, statement-1 is true.
We have,
$|\vec a +\vec b|^2 = |\vec a|^2 + |\vec b|^2+2|\vec a||\vec b|\cos θ$
So, $|\vec a +\vec b|$ is greatest or least according as $\cos θ$ is greatest or least.
Hence, the greatest and least values of $|\vec a +\vec b|$ bare given by
$|\vec a +\vec b|^2 = |\vec a|^2 + |\vec b|^2+2|\vec a||\vec b|=(|\vec a|+|\vec b|)^2$
and, $|\vec a -\vec b|^2 = |\vec a|^2 + |\vec b|^2-2|\vec a||\vec b|=(|\vec a|-|\vec b|)^2$
∴ Greatest value of $|\vec a +\vec b|=|\vec a|+|\vec b|$
Least value of $|\vec a +\vec b|=\left||\vec a|-|\vec b|\right|$