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  • If <img alt="\overrightarrow{\mathrm{a}}=\hat{\imath}+2 \hat{k}, \vec{b}=\hat{\imath}+\hat{\jmath}+\hat{k}, \vec{c}=7 \hat{\imath}-3 \hat{\jmath}+4 \hat{k}, \overrightarrow{\mathrm{r}} \times \over

If \overrightarrow{\mathrm{a}}=\hat{\imath}+2 \hat{k}, \vec{b}=\hat{\imath}+\hat{\jmath}+\hat{k}, \vec{c}=7 \hat{\imath}-3 \hat{\jmath}+4 \hat{k}, \overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{0} and  \overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{a}}=0.
Then \vec{r} \cdot \vec{c} is equal to

Option: 1

32


Option: 2

30


Option: 3

36


Option: 4

34


Answers (1)

best_answer

\begin{aligned} & \overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=0 \\ & \Rightarrow \overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}}=0 \\ & \Rightarrow(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{c}}) \times \overrightarrow{\mathrm{b}}=0 \\ & \overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{c}} \| \overrightarrow{\mathrm{b}} \\ & \overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{c}}=\lambda \overrightarrow{\mathrm{b}} \\ & \overrightarrow{\mathrm{r}}=\lambda \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}} \end{aligned}

\begin{aligned} & =\lambda(\mathrm{i}+\mathrm{j}+\mathrm{k})+(7 \mathrm{i}-3 \mathrm{j}+4 \mathrm{k}) \\ & =\mathrm{i}(\lambda+7)+\mathrm{j}(\lambda-3)+\mathrm{k}(\lambda+4) \\ & \overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{a}}=0 \\ & \Rightarrow(7+\lambda)+2(\lambda+4)=0 \\ & \Rightarrow 3 \lambda=-15 \Rightarrow \lambda=-5 \end{aligned}

\begin{aligned} & \therefore \overrightarrow{\mathrm{r}}=2 \mathrm{i}-8 \mathrm{j}-\mathrm{k} \\ & \overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{c}}=(2 \mathrm{i}-8 \mathrm{j}-\mathrm{k}) \cdot(7 \mathrm{i}-3 \mathrm{j}+4 \mathrm{k}) \\ & =14+24-4=34 \end{aligned}

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Gaurav

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