Question - Determining Perpendicular Vectors by Dot Product
Solution:
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.
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