QJEEM17Ma90bb: JEE Main 2017 Physics - Class 11 JEEM17Ma90bb Solution: If the vector is written as the sum of a vector paralel to and…

Question

Verified Solution • 2025–26 Session moderate

If the vector

\overrightarrow b = 3\widehat j + 4\widehat k

is written as the sum of a vector

\overrightarrow {{b_1}} ,

paralel to

\overrightarrow a = \widehat i + \widehat j

and a vector

\overrightarrow {{b_2}} ,

perpendicular to

\overrightarrow a ,

then

\overrightarrow {{b_1}} \times \overrightarrow {{b_2}}

is equal to :

Accepted Answer

\overrightarrow {{b_1}} = {{\left( {\overrightarrow {{b_1}} .\overrightarrow a } \right)\widehat a} \over 1}

=

\left\{ {{{\left( {3\widehat j + 4\widehat k} \right).\left( {\widehat i + \widehat j} \right)} \over {\sqrt 2 }}} \right\}\left( {{{\widehat i + \widehat j} \over {\sqrt 2 }}} \right)

=

{{3\left( {\widehat i + \widehat j} \right)} \over {\sqrt 2 \times \sqrt 2 }} = {{3\left( {\widehat i + \widehat j} \right)} \over 2}

\overrightarrow {{b_1}} + \overrightarrow {{b_2}} = \overrightarrow b

\Rightarrow

\overrightarrow {{b_2}} = \overrightarrow b - \overrightarrow {{b_1}}

=

\left( {3\widehat j + 4\widehat k} \right) - {3 \over 2}\left( {\widehat i + \widehat j} \right)

\Rightarrow

\overrightarrow {{b_2}}

=

- {3 \over 2}\widehat i + {3 \over 2}\widehat j + 4\widehat k

&

\vec{b_{1}} \times \vec{b_{2}} = | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{3}{2} & \frac{3}{2} & 0 \\ - \frac{3}{2} & \frac{3}{2} & 4 \end{matrix} |

\Rightarrow

\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} = \widehat i\left( 6 \right) - \widehat j\left( 6 \right) + \widehat k\left( { - {9 \over 4} + {9 \over 4}} \right)

\Rightarrow

6\widehat i - 6\widehat j + {9 \over 2}\widehat k

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