Given the vectors ( vec{A} = 2vec{i} - 3vec{j} + k ) and ( vec{B} = -vec{i} + 2vec{j} + 4k ). If ( vec{A} ) and ( vec{B} ) are orthogonal to each other, find the value of ( k ).

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Step:1 ：Step 1: Since vectors ( vec{A} ) and ( vec{B} ) are orthogonal, their dot product will be equal to zero. That is, ( vec{A}cdotvec{B} = 0 ).Step:2 ：Step 2: Calculating the dot product, we get ( vec{A}cdotvec{B} = (2)(-1) + (-3)(2) + (1)(4k) ).Step:3 ：Step 3: Simplifying, we get ( -2 -6 + 4k = 0 ).Step:4 ：Step 4: Solving for ( k ), we get ( 4k = 8 ) and finally, ( k = 2 ).

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Step: 1 ：Step 1: Since vectors ( vec{A} ) and ( vec{B} ) are orthogonal, their dot product will be equal to zero. That is, ( vec{A}cdotvec{B} = 0 ).Step: 2 ：Step 2: Calculating the dot product, we get ( vec{A}cdotvec{B} = (2)(-1) + (-3)(2) + (1)(4k) ).Step: 3 ：Step 3: Simplifying, we get ( -2 -6 + 4k = 0 ).Step: 4 ：Step 4: Solving for ( k ), we get ( 4k = 8 ) and finally, ( k = 2 ).

Given the vectors \( \vec{A} = 2\vec{i} - 3\vec{j} + k \) and \( \vec{B} = -\vec{i} + 2\vec{j} + 4k \). If \( \vec{A} \) and \( \vec{B} \) are orthogonal to each other, find the value of \( k \).

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