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(C) Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + \hat{j}$.$A$ vector perpendicular to both $\vec{a}$ and $\vec{b}$ is given by

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$c(\hat{i} - \hat{j})$,where $c$ is a scalar

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(C) Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + \hat{j}$.$A$ vector perpendicular to both $\vec{a}$ and $\vec{b}$ is given by their cross product $\vec{a} 	imes \vec{b}$.$\vec{a} 	imes \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 1 & 1 & 1 \ 1 & 1 & 0 \end{vmatrix}$$= \hat{i}(0 - 1) - \hat{j}(0 - 1) + \hat{k}(1 - 1)$$= -\hat{i} + \hat{j} + 0\hat{k} = -(\hat{i} - \hat{j})$.Any vector perpendicular to both is a scalar multiple of this vector,which can be written as $c(\hat{i} - \hat{j})$ for some scalar $c$.

Find a vector that is perpendicular to both vectors $\hat{i} + \hat{j} + \hat{k}$ and $\hat{i} + \hat{j}$.

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