Mathematics

Problem

\mathbf{a} = \hat{\mathbf{i}} + \hat{\mathbf{j}} + \hat{\mathbf{k}}

and

\mathbf{b} = \hat{\mathbf{i}} - \hat{\mathbf{j}}

, then the vectors

(\mathbf{a} \cdot \hat{\mathbf{i}})\hat{\mathbf{i}} + (\mathbf{a} \cdot \hat{\mathbf{j}})\hat{\mathbf{j}} + (\mathbf{a} \cdot \hat{\mathbf{k}})\hat{\mathbf{k}}

(\mathbf{b} \cdot \hat{\mathbf{i}})\hat{\mathbf{i}} + (\mathbf{b} \cdot \hat{\mathbf{j}})\hat{\mathbf{j}} + (\mathbf{b} \cdot \hat{\mathbf{k}})\hat{\mathbf{k}}

and

\hat{\mathbf{i}} + \hat{\mathbf{j}} - 2\hat{\mathbf{k}}

a. are mutually perpendicular b. are coplanar c. form a parallelopiped of volume 6 units d. form a parallelopiped of volume 3 units