# Find $\mid\overrightarrow{x}\mid$,if for a unit vector $\overrightarrow{a},(\overrightarrow{x}-\overrightarrow{a}).(\overrightarrow{x}+\overrightarrow{a})=15.$

This question appeared in 65-1,65-2 and 65-3 versions of the paper in 2013.

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• Given a unit vector $\overrightarrow{a}$, the magnitude of a unit vector $=\mid\overrightarrow{a}\mid=1$
Given $(\overrightarrow{x}-\overrightarrow{a}).(\overrightarrow{x}+\overrightarrow{a})=15.$
$(\overrightarrow{x}-\overrightarrow{a}).(\overrightarrow{x}+\overrightarrow{a}) = \overrightarrow{x}.\overrightarrow{x} + \overrightarrow{x}. \overrightarrow{a} - \overrightarrow{a}. \overrightarrow{x} + \overrightarrow{a}\overrightarrow{a}$$= {\mid\overrightarrow{x}\mid}^2-{\mid\overrightarrow{a}\mid}^2$
$\Rightarrow {\mid\overrightarrow{x}\mid}^2-{\mid\overrightarrow{a}\mid}^2=15.$
Given a unit vector $\overrightarrow{a}$, the magnitude of a unit vector $=\mid\overrightarrow{a}\mid=1$
$\Rightarrow {\mid\overrightarrow{x}\mid}^2-1=15.$
$\Rightarrow {\mid\overrightarrow{x}\mid}^2=16.$
$\Rightarrow\; \mid\overrightarrow{x}\mid=4.$
edited Mar 21, 2013