22. For what value of \( t \) is the vector \( 2 t \boldsymbol{i}+4 \boldsymbol{j}-(10+t) \boldsymbol{k} \) perpendicular to the vector \( \boldsymbol{i}+t \boldsymbol{j}+\boldsymbol{k} \) ?

To determine the value of \( t \) for which the vectors \( \mathbf{A} = 2t \mathbf{i} + 4 \mathbf{j} - (10 + t) \mathbf{k} \) and \( \mathbf{B} = \mathbf{i} + t \mathbf{j} + \mathbf{k} \) are perpendicular, we can use the dot product. Two vectors are perpendicular if their dot product is zero. Let's compute the dot product: \[ \mathbf{A} \cdot \mathbf{B} = (2t)(1) + (4)(t) + [-(10 + t)(1)] \] Expanding this gives us: \[ \mathbf{A} \cdot \mathbf{B} = 2t + 4t - (10 + t) = 2t + 4t - 10 - t = (2t + 4t - t - 10) = (5t - 10) \] Setting the dot product to zero for perpendicularity: \[ 5t - 10 = 0 \] Solving for \( t \): \[ 5t = 10 \\ t = 2 \] Thus, the value of \( t \) for which the vector \( 2t \boldsymbol{i} + 4 \boldsymbol{j} - (10 + t) \boldsymbol{k} \) is perpendicular to the vector \( \boldsymbol{i} + t \boldsymbol{j} + \boldsymbol{k} \) is \( t = 2 \).