QJEEM20Ma628f: JEE Main 2020 Physics - Class 11 JEEM20Ma628f Solution: Let the vectors , , be such that , and . If the projection of…

Question

Verified Solution • 2025–26 Session moderate

Let the vectors

\overrightarrow a

,

\overrightarrow b

,

\overrightarrow c

be such that

\left| {\overrightarrow a } \right| = 2

,

\left| {\overrightarrow b } \right| = 4

and

\left| {\overrightarrow c } \right| = 4

. If the projection of

\overrightarrow b

on

\overrightarrow a

is equal to the projection of

\overrightarrow c

on

\overrightarrow a

and

\overrightarrow b

is perpendicular to

\overrightarrow c

, then the value of

\left| {\overrightarrow a + \vec b - \overrightarrow c } \right|

is ___________.

Accepted Answer

Projection of

\overrightarrow b

on

\overrightarrow a

= Projection of

\overrightarrow c

on

\overrightarrow a

\Rightarrow

{{\overrightarrow b .\overrightarrow a } \over {\left| {\overrightarrow a } \right|}} = {{\overrightarrow c .\overrightarrow a } \over {\left| {\overrightarrow a } \right|}}

\Rightarrow

\overrightarrow b .\overrightarrow a = \overrightarrow c .\overrightarrow a

\because

\overrightarrow b

is perpendicular to

\overrightarrow c

\therefore

\overrightarrow b .\overrightarrow c = 0

Let

\left| {\overrightarrow a + \vec b - \overrightarrow c } \right|

= k

Square both sides

k² =

{{{\left( {\overrightarrow a } \right)}^2}}

+

{{{\left( {\overrightarrow b } \right)}^2}}

+

{{{\left( {\overrightarrow c } \right)}^2}}

+

2\overrightarrow a .\overrightarrow b

-

2\overrightarrow b .\overrightarrow c

-

2\overrightarrow a .\overrightarrow c

\Rightarrow

k² =

{{{\left( {\overrightarrow a } \right)}^2}}

+

{{{\left( {\overrightarrow b } \right)}^2}}

+

{{{\left( {\overrightarrow c } \right)}^2}}

\Rightarrow

k² = 2² + 4² + 4² = 36

\Rightarrow

k = 6

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