Finding the Value of P for Perpendicular Vectors
To determine the value of \( P \) such that the vectors \( \mathbf{A} = 2\mathbf{i} - 3\mathbf{j} + p\mathbf{k} \) and \( \mathbf{B} = 3\mathbf{i} - 4\mathbf{j} + \mathbf{k} \) are mutually perpendicular, we use the dot product. Two vectors are perpendicular if their dot product equals zero.
The dot product \( \mathbf{A} \cdot \mathbf{B} \) is calculated as follows:
\[
\mathbf{A} \cdot \mathbf{B} = (2)(3) + (-3)(-4) + (p)(1) = 6 + 12 + p = 18 + p
\]
Setting the dot product to zero for perpendicularity:
\[
18 + p = 0
\]
Solving for \( P \):
\[
p = -18
\]
Thus, the value of \( P \) is \( -18 \).
Proving Vectors A and B are Parallel
To prove that the vectors \( \mathbf{A} = \mathbf{i} - 2\mathbf{j} + \mathbf{k} \) and \( \mathbf{B} = 2\mathbf{i} - 4\mathbf{j} + 2\mathbf{k} \) are parallel, we need to check if one vector is a scalar multiple of the other.
1. Scalar Multiple Check:
- We can express \( \mathbf{B} \) in terms of \( \mathbf{A} \):
- Notice that every component of \( \mathbf{B} \) can be related to \( \mathbf{A} \):
\[
\mathbf{B} = 2\mathbf{A}
\]
- This implies:
\[
2(\mathbf{i} - 2\mathbf{j} + \mathbf{k}) = 2\mathbf{i} - 4\mathbf{j} + 2\mathbf{k}
\]
2. Conclusion:
- Since \( \mathbf{B} \) is a scalar multiple of \( \mathbf{A} \) (specifically, \( 2 \)), we conclude that the vectors \( \mathbf{A} \) and \( \mathbf{B} \) are indeed parallel.
Thus, \( \mathbf{A} \) and \( \mathbf{B} \) are parallel vectors.