A) \[\frac{{{\pi }^{2}}}{4}\,m{{s}^{-\,2}}\] and direction along the radius towards the centre
B) \[{{\pi }^{2}}\,m{{s}^{-}}^{2}\] and direction along the radius away from centre
C) \[{{\pi }^{2}}\,m{{s}^{-}}^{2}\] and direction along the radius towards the centre
D) \[{{\pi }^{2}}\,m{{s}^{-}}^{2}\] and direction along the tangent to the circle
Correct Answer: C
Solution :
We are given \[\overrightarrow{a}=2\,\hat{i}+3\,\hat{j}+8\,\hat{k},\,\,\vec{b}=4\,\hat{j}-4\,\hat{i}+\alpha \hat{k}\] \[= -4\hat{i} + 4\hat{j} +a\hat{k}\] According to the above hypothesis, If \[\vec{a}\,\,\bot \text{ }\vec{b}\], then dot product of vectors should be zero \[\hat{a}\,\,.\,\,\hat{b}=0\] \[\Rightarrow \,\,\,\,(2\hat{i}+3\hat{j}+8\hat{k})\,(-\,4\hat{i}+4\hat{j}+a\hat{k})=0\] \[\Rightarrow \,\,\,-8+12+8\alpha \,\,=\,\,0\,\,\,\,\Rightarrow \,\, 8\alpha =-\,4\] \[\therefore \,\,\,\,\,\,\,\,\,\,\,a=-\frac{4}{8}\,\,=\,\,-\frac{1}{2}\]
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