A) Lie on the plane
B) Are on the same side of the plane
C) Are on the opposite side of the plane
D) None of these
Correct Answer: C
Solution :
The position vectors of two given points are \[\mathbf{a}=\mathbf{i}-\mathbf{j}+3\mathbf{k}\] and \[\mathbf{b}=3\mathbf{i}+3\mathbf{j}+3\mathbf{k}\] the equation of the given plane is \[\mathbf{r}.(5\mathbf{i}+2\mathbf{j}-7\mathbf{k})+9=0\] or \[\mathbf{r}.\,\mathbf{n}+d=0\]. We have, \[\mathbf{a}.\,\mathbf{n}+d=(\mathbf{i}-\mathbf{j}+3\mathbf{k}).(5\mathbf{i}+2\mathbf{j}-7\mathbf{k})+9\] \[=5-2-21+9<0\] and, \[\mathbf{b}.\mathbf{n}+d=(3\mathbf{i}+3\mathbf{j}+3\mathbf{k}).(5\mathbf{i}+2\mathbf{j}-7\mathbf{k})+9\] \[=15+6-21+9>0\] So, the points a and \[\mathbf{b}\]are on the opposite sides of the plane.
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