A) \[\overrightarrow{r}.[\overrightarrow{r}-(2\hat{i}+\hat{j}+\hat{k})]=\frac{2}{5}\]
B) \[\overrightarrow{r}.[\overrightarrow{r}-(2\hat{i}-3\hat{j}-4\hat{k})]=\frac{1}{2}\]
C) \[\overrightarrow{r}.[\overrightarrow{r}-(2\hat{i}+3\hat{j}+4\hat{k})]=\frac{5}{2}\]
D) \[\overrightarrow{r}.[\overrightarrow{r}+(2\hat{i}-3\hat{j}-4\hat{k})]=\frac{5}{2}\]
E) \[\overrightarrow{r}.[\overrightarrow{r}-(2\hat{i}-3\hat{j}-4\hat{k})]=\frac{5}{2}\]
Correct Answer: E
Solution :
Given equation can be rewritten as \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-2x+3y+4z-\frac{5}{2}=0\] Let \[\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}\] \[\therefore \]Given equation written in vector form is \[\overrightarrow{r}.[\overrightarrow{r}-(2\hat{i}-3\hat{j}-4\hat{k})]=\frac{5}{2}\]
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