Let vec{a}=hat{i}+hat{j}+2 hat{k} and vec{b}= | vedclass

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(D) Given $\vec{a}=\hat{i}+\hat{j}+2 \hat{k}$ and $\vec{b}=-\hat{i}+2 \hat{j}+3 \hat{k}$.First,calculate $\vec{a}+\vec{b} = (1-1)\hat{i} + (1+2)\hat{j

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$7(34 \hat{i}-5 \hat{j}+3 \hat{k})$

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(D) Given $\vec{a}=\hat{i}+\hat{j}+2 \hat{k}$ and $\vec{b}=-\hat{i}+2 \hat{j}+3 \hat{k}$.First,calculate $\vec{a}+\vec{b} = (1-1)\hat{i} + (1+2)\hat{j} + (2+3)\hat{k} = 3\hat{j} + 5\hat{k}$.Next,simplify the expression $E = ((\vec{a} 	imes((\vec{a}-\vec{b}) 	imes \vec{b})) 	imes \vec{b})$.Using the property $(\vec{a}-\vec{b}) 	imes \vec{b} = \vec{a} 	imes \vec{b} - \vec{b} 	imes \vec{b} = \vec{a} 	imes \vec{b}$.So,$E = ((\vec{a} 	imes (\vec{a} 	imes \vec{b})) 	imes \vec{b})$.Using the vector triple product formula $\vec{u} 	imes (\vec{v} 	imes \vec{w}) = (\vec{u} \cdot \vec{w})\vec{v} - (\vec{u} \cdot \vec{v})\vec{w}$,we have:$\vec{a} 	imes (\vec{a} 	imes \vec{b}) = (\vec{a} \cdot \vec{b})\vec{a} - (\vec{a} \cdot \vec{a})\vec{b}$.Then $E = ((\vec{a} \cdot \vec{b})\vec{a} - (\vec{a} \cdot \vec{a})\vec{b}) 	imes \vec{b} = (\vec{a} \cdot \vec{b})(\vec{a} 	imes \vec{b}) - (\vec{a} \cdot \vec{a})(\vec{b} 	imes \vec{b}) = (\vec{a} \cdot \vec{b})(\vec{a} 	imes \vec{b})$.Calculate $\vec{a} \cdot \vec{b} = (1)(-1) + (1)(2) + (2)(3) = -1 + 2 + 6 = 7$.Calculate $\vec{a} 	imes \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 1 & 1 & 2 \ -1 & 2 & 3 \end{vmatrix} = \hat{i}(3-4) - \hat{j}(3+2) + \hat{k}(2+1) = -\hat{i} - 5\hat{j} + 3\hat{k}$.Thus,$E = 7(-\hat{i} - 5\hat{j} + 3\hat{k})$.Finally,calculate $(\vec{a}+\vec{b}) 	imes E = (3\hat{j} + 5\hat{k}) 	imes 7(-\hat{i} - 5\hat{j} + 3\hat{k}) = 7 \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 0 & 3 & 5 \ -1 & -5 & 3 \end{vmatrix}$.$= 7 [\hat{i}(9 - (-25)) - \hat{j}(0 - (-5)) + \hat{k}(0 - (-3))] = 7(34\hat{i} - 5\hat{j} + 3\hat{k})$.

Let $\vec{a}=\hat{i}+\hat{j}+2 \hat{k}$ and $\vec{b}=-\hat{i}+2 \hat{j}+3 \hat{k}$. Then the vector product $(\vec{a}+\vec{b}) \times((\vec{a} \times((\vec{a}-\vec{b}) \times \vec{b})) \times \vec{b})$ is equal to:

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