A) \[-1\]
B) \[\frac{1}{2}\]
C) \[-\frac{1}{2}\]
D) \[1\]
Correct Answer: C
Solution :
Key Idea : Two vectors must be perpendicular if their dot product is zero. Let \[\overrightarrow{a}=2\hat{i}+3\hat{j}+8\hat{k}\] \[\overrightarrow{b}=4\hat{i}-4\hat{j}+\alpha \hat{k}\] \[=-\,4\hat{i}+4\hat{j}+\alpha \hat{k}\] According to the above hypothesis: \[\overrightarrow{a}\bot \overrightarrow{b}\] \[\Rightarrow \] \[\overrightarrow{a}\cdot \overrightarrow{b}=0\] \[\Rightarrow \] \[(2\hat{i}+3\hat{j}+8\hat{k})\cdot (-\,4\hat{i}+4\hat{j}+\alpha \hat{k})=0\] \[\Rightarrow \] \[-\,8+12+8\alpha =0\] \[\Rightarrow \] \[8\alpha =-\,4\] \[\therefore \] \[\alpha =-\frac{4}{8}=-\frac{1}{2}\] NOTE: \[\overrightarrow{a}\cdot \overrightarrow{b}=ab\,\,\cos \theta .\] Here, a and b are always positive as they are the magnitudes of \[\overrightarrow{a}\]and \[\overrightarrow{b}\].
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