Determining Perpendicular Vectors by Dot Product
To determine whether two vectors are perpendicular, their dot product must equal zero.
Given the vectors \( \vec{A} = k\hat{i} + 4\hat{j} \) and \( \vec{B} = -\hat{i} + 2\hat{j} \), the dot product of these two vectors is computed as:
\( \vec{A} \cdot \vec{B} = (k\hat{i} + 4\hat{j}) \cdot (-\hat{i} + 2\hat{j}) \)
This results in:
\( \vec{A} \cdot \vec{B} = k(-1) + 4(2) \)
\( \vec{A} \cdot \vec{B} = -k + 8 \)
Since the vectors are perpendicular, their dot product is zero. Therefore:
\( -k + 8 = 0 \)
Solving for \( k \) gives us:
\( k = 8 \)
So the value of \( k \) is 8, which corresponds to option D.
