QJEEM23Ma4dbc: JEE Main 2023 Physics - Class 11 JEEM23Ma4dbc Solution: Let and be three vectors. If is a vector such that, and , then…

Question

Verified Solution • 2025–26 Session moderate

Let

\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=\hat{i}-\hat{j}+2 \hat{k}

and

\vec{c}=5 \hat{i}-3 \hat{j}+3 \hat{k}

be three vectors. If

\vec{r}

is a vector such that,

\vec{r} \times \vec{b}=\vec{c} \times \vec{b}

and

\vec{r} \cdot \vec{a}=0

, then

25|\vec{r}|^{2}

is equal to :

Accepted Answer

\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}

\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}

\overrightarrow{\mathrm{c}}=\hat{5 \mathrm{i}}-3 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}

(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{c}}) \times \vec{b}=0, \quad \vec{r} \cdot \vec{a}=0

\Rightarrow \vec{r}-\vec{c}=\lambda \vec{b}

Also,

(\vec{c}+\lambda \vec{b}) \cdot \vec{a}=0

\Rightarrow \vec{a} \cdot \vec{c}+\lambda(\vec{a} \cdot \vec{b})=0

\therefore \lambda=\frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}}=\frac{-8}{5}

\overrightarrow{\mathrm{r}}=\frac{5(5 \hat{\mathrm{i}}-3 \hat{\mathrm{i}}+3 \hat{\mathrm{k}})-8(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})}{5}

\Rightarrow

\overrightarrow{\mathrm{r}}=\frac{17 \hat{\mathrm{i}}-7 \hat{\mathrm{j}}+\hat{\mathrm{k}}}{5}

\Rightarrow

|\overrightarrow{\mathrm{r}}|^{2}=\frac{1}{25}(289+50)

\Rightarrow

25|\overrightarrow{\mathrm{r}}|^{2}=339

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