If the angle between the vectors `x hat i+3 hat j-7 hat k\ a n d\ x hat i-x hat j+4 hat k` acute, then `x` lies in the interval `(-4,7)` b. `[-4,\ 7]` c. `R-[-4,7]` d. `R-(4,7)`

Explanation

To solve the problem, we need to determine the conditions under which the angle between the two vectors is acute. The vectors given are:

1. \( \mathbf{A} = \hat{i} + 3\hat{j} - 7\hat{k} \)
2. \( \mathbf{B} = \hat{i} - \hat{j} + 4\hat{k} \)

### Step 1: Calculate the Dot Product
The dot product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by:

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