If \( P(\vec{r}) \) is a position vector of a point with scalar components \( (x,y,z) \), then the value of \( (\vec{r}.\hat{\imath})(-\hat{\imath})-(\vec{r}.\hat{\jmath})\hat{\jmath}+(\vec{r}.(-\hat{k}))\hat{k} \), is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( -(x\hat{\imath}+y\hat{\jmath}+z\hat{k}) \)
With \( \vec{r}=x\hat{\imath}+y\hat{\jmath}+z\hat{k} \), the dot products with the unit vectors give one component each.
\( \displaystyle \vec{r}.\hat{\imath}=x \)
\( \displaystyle \vec{r}.\hat{\jmath}=y \)
\( \displaystyle \vec{r}.(-\hat{k})=-z \)
Substituting in the expression,
\( \displaystyle (\vec{r}.\hat{\imath})(-\hat{\imath})-(\vec{r}.\hat{\jmath})\hat{\jmath}+(\vec{r}.(-\hat{k}))\hat{k} \)
\( \displaystyle =-x\hat{\imath}-y\hat{\jmath}-z\hat{k} \)
\( \displaystyle =-(x\hat{\imath}+y\hat{\jmath}+z\hat{k}) \)
The result is a vector, not a scalar.