Mathematics - Vector Question with Solution | TestHub

$\overrightarrow{r}\times \overrightarrow{a}-\overrightarrow{r}\times \overrightarrow{b}=0$

$\Rightarrow \overrightarrow{r}\times \left(\overrightarrow{a}-\overrightarrow{b}\right)=0$

$\Rightarrow \overrightarrow{r}=\lambda \left(\overrightarrow{a}-\overrightarrow{b}\right)$

$\Rightarrow \overrightarrow{r}=\lambda \left(-5\hat{i}-4\hat{j}+10\hat{k}\right)$

Also $\overrightarrow{r}·\left(\hat{i}+2\hat{j}+\hat{k}\right)=-3$

$\Rightarrow \lambda \left(-5-8+10\right)=-3$

$\lambda =1$

Now $\overrightarrow{r}=-5\hat{i}-4\hat{j}+10\hat{k}$

$=\overrightarrow{r}·\left(2\hat{i}-3\hat{j}+\hat{k}\right)$

$=-10+12+10=12$