The vectors `2 hat i+3 hat j-4 hat k\ a n d\ a hat i=b hat j+c hat k` are perpendicular if `a=2,\ b=3,\ c=-4` `a=4,\ b=4,\ c=5` `a=4,\ b=4,\ c=-5` `a=-4,\ b=4,\ c=-5`

Explanation

To determine the values of \( a \), \( b \), and \( c \) for which the vectors \( \mathbf{u} = 2\hat{i} + 3\hat{j} - 4\hat{k} \) and \( \mathbf{v} = a\hat{i} + b\hat{j} + c\hat{k} \) are perpendicular, we can use the property that the dot product of two perpendicular vectors is zero.

### Step-by-Step Solution:

1. **Write the dot product equation:**
The dot product of vectors \( \mathbf{u} \) and \( \mathbf{v} \) is given by:
\[
\mathbf{u} \cdot \mathbf{v} = (2\hat{i} + 3\hat{j} - 4\hat{k}) \cdot (a\hat{i} + b\hat{j} + c\hat{k})
...