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# If the Sum of Two Unit Vectors is a Unit Vector Prove that the Magnitude of Their Difference is √ 3 . - CBSE (Science) Class 12 - Mathematics

ConceptBasic Concepts of Vector Algebra

#### Question

If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is $\sqrt{3}$.

#### Solution

$\text{ Given that } \hat{a}\ , \hat{b}\ \text{ and }\left| \hat{a} + \hat{b} \right|\text{ are unit vectors }.$
$So,\left| \hat{a} \right|=1,\left| \hat{b} \right|=1 and \left| \left| \hat{a} + \hat{b} \right| \right|=1$
$\text{ We have }$
$\left| \hat{a} + \hat{b} \right|^2 + \left| \hat{a} - \hat{b} \right|^2 = 2\left( \left| \hat{a} \right|^2 + \left| \hat{b} \right|^2 \right)$
$\Rightarrow 1 + \left| \hat{a} - \hat{b} \right|^2 = 2\left( 1 + 1 \right)$
$\Rightarrow 1 + \left| \hat{a} - \hat{b} \right|^2 = 4$
$\Rightarrow \left| \hat{a} - \hat{b} \right|^2 = 3$
$\Rightarrow \left| \hat{a} - \hat{b} \right| = \sqrt{3}$

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Solution If the Sum of Two Unit Vectors is a Unit Vector Prove that the Magnitude of Their Difference is √ 3 . Concept: Basic Concepts of Vector Algebra.
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