Three vectors vec{a} = hat{i} + hat{j},vec{b} | vedclass

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(D) Let $\vec{a} = \hat{i} + \hat{j}$,$\vec{b} = \hat{j} + \hat{k}$,and $\vec{c} = \hat{k} + \hat{i}$.The unit vectors perpendicular to the planes are

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$\frac{4}{3\sqrt{3}}$ cubic units

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(D) Let $\vec{a} = \hat{i} + \hat{j}$,$\vec{b} = \hat{j} + \hat{k}$,and $\vec{c} = \hat{k} + \hat{i}$.The unit vectors perpendicular to the planes are given by:$\vec{\alpha} = \frac{\vec{a} 	imes \vec{b}}{|\vec{a} 	imes \vec{b}|}$,$\vec{\beta} = \frac{\vec{b} 	imes \vec{c}}{|\vec{b} 	imes \vec{c}|}$,and $\vec{\gamma} = \frac{\vec{c} 	imes \vec{a}}{|\vec{c} 	imes \vec{a}|}$.The volume of the parallelepiped formed by these vectors is the scalar triple product $|[\vec{\alpha}, \vec{\beta}, \vec{\gamma}]|$.$|[\vec{\alpha}, \vec{\beta}, \vec{\gamma}]| = \frac{|[\vec{a} 	imes \vec{b}, \vec{b} 	imes \vec{c}, \vec{c} 	imes \vec{a}]|}{|\vec{a} 	imes \vec{b}| |\vec{b} 	imes \vec{c}| |\vec{c} 	imes \vec{a}|} = \frac{[\vec{a}, \vec{b}, \vec{c}]^2}{|\vec{a} 	imes \vec{b}| |\vec{b} 	imes \vec{c}| |\vec{c} 	imes \vec{a}|}$.First,calculate the scalar triple product $[\vec{a}, \vec{b}, \vec{c}]$:$[\vec{a}, \vec{b}, \vec{c}] = \begin{vmatrix} 1 & 1 & 0 \ 0 & 1 & 1 \ 1 & 0 & 1 \end{vmatrix} = 1(1-0) - 1(0-1) + 0 = 1 + 1 = 2$.Next,calculate the cross products:$\vec{a} 	imes \vec{b} = (\hat{i} + \hat{j}) 	imes (\hat{j} + \hat{k}) = \hat{k} - \hat{i} + \hat{j} = \hat{i} + \hat{j} + \hat{k}$.$\vec{b} 	imes \vec{c} = (\hat{j} + \hat{k}) 	imes (\hat{k} + \hat{i}) = \hat{i} + \hat{j} + \hat{k}$.$\vec{c} 	imes \vec{a} = (\hat{k} + \hat{i}) 	imes (\hat{i} + \hat{j}) = \hat{j} - \hat{k} + \hat{i} = \hat{i} + \hat{j} + \hat{k}$.The magnitude of each cross product is $|\vec{a} 	imes \vec{b}| = |\vec{b} 	imes \vec{c}| = |\vec{c} 	imes \vec{a}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}$.Therefore,the volume is $\frac{2^2}{\sqrt{3} \cdot \sqrt{3} \cdot \sqrt{3}} = \frac{4}{3\sqrt{3}}$ cubic units.

Three vectors $\vec{a} = \hat{i} + \hat{j}$,$\vec{b} = \hat{j} + \hat{k}$,and $\vec{c} = \hat{k} + \hat{i}$ are given. If three unit vectors are drawn perpendicular to the three planes formed by these vectors,what is the volume of the parallelepiped formed by these unit vectors?

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