Find the value of `c` for which the vectors ` vec a=(clog_2x) hat i-6 hat j+3 hat k` and `vec b=((log)_2x) hat i+2 hat j+(2c(log)_2x) hat k` make an obtuse angle for any `x in (0,oo)` .

Explanation

To find the value of \( c \) for which the vectors \( \vec{a} = (c \log_2 x) \hat{i} - 6 \hat{j} + 3 \hat{k} \) and \( \vec{b} = (\log_2 x) \hat{i} + 2 \hat{j} + (2c \log_2 x) \hat{k} \) make an obtuse angle for any \( x \in (0, \infty) \), we can follow these steps:

### Step 1: Understand the Condition for Obtuse Angle
The angle between two vectors \( \vec{a} \) and \( \vec{b} \) is obtuse if their dot product is less than zero:
\[ \vec{a} \cdot \vec{b} < 0 \]

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