QJEEM20Ma5577: JEE Main 2020 Physics - Class 11 JEEM20Ma5577 Solution: Let , and be three unit vectors such that + = 8. Then + is…

Question

Verified Solution • 2025–26 Session moderate

Let

\overrightarrow a

,

\overrightarrow b

and

\overrightarrow c

be three unit vectors such that

{\left| {\overrightarrow a - \overrightarrow b } \right|^2}

+

{\left| {\overrightarrow a - \overrightarrow c } \right|^2}

= 8.

Then

{\left| {\overrightarrow a + 2\overrightarrow b } \right|^2}

+

{\left| {\overrightarrow a + 2\overrightarrow c } \right|^2}

is equal to ______.

Accepted Answer

Given,

\left| {\overrightarrow a } \right| = \left| {\overrightarrow b } \right| = \left| {\overrightarrow c } \right| = 1

{\left| {\overrightarrow a - \overrightarrow b } \right|^2}

+

{\left| {\overrightarrow a - \overrightarrow c } \right|^2}

= 8

\Rightarrow

{\left| {\overrightarrow a } \right|^2} + {\left| {\overrightarrow b } \right|^2} - 2\overrightarrow a .\overrightarrow b + {\left| {\overrightarrow a } \right|^2} + {\left| {\overrightarrow c } \right|^2} - 2\overrightarrow a .\overrightarrow c

= 8

\Rightarrow

\overrightarrow a .\overrightarrow b + \overrightarrow a .\overrightarrow c

= -2

Now,

{\left| {\overrightarrow a + 2\overrightarrow b } \right|^2}

+

{\left| {\overrightarrow a + 2\overrightarrow c } \right|^2}

=

{\left| {\overrightarrow a } \right|^2} + 4{\left| {\overrightarrow b } \right|^2} + 4\overrightarrow a .\overrightarrow b + {\left| {\overrightarrow a } \right|^2} + 4{\left| {\overrightarrow c } \right|^2} + 4\overrightarrow a .\overrightarrow c

= 10 + 4

\left( {\overrightarrow a .\overrightarrow b + \overrightarrow a .\overrightarrow c } \right)

= 10 + 4(-2)

= 2

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