If overline{a}=frac{1}{sqrt{10}}(3 hat{i}+hat | vedclass

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(D) First,we calculate the magnitudes of $\overline{a}$ and $\overline{b}$:$|\overline{a}| = \frac{1}{\sqrt{10}} \sqrt{3^2 + 0^2 + 1^2} = \frac{\sqrt{

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$-5$

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(D) First,we calculate the magnitudes of $\overline{a}$ and $\overline{b}$:$|\overline{a}| = \frac{1}{\sqrt{10}} \sqrt{3^2 + 0^2 + 1^2} = \frac{\sqrt{10}}{\sqrt{10}} = 1$$|\overline{b}| = \frac{1}{7} \sqrt{2^2 + 3^2 + (-6)^2} = \frac{1}{7} \sqrt{4 + 9 + 36} = \frac{7}{7} = 1$Next,we calculate the dot product $\overline{a} \cdot \overline{b}$:$\overline{a} \cdot \overline{b} = \frac{1}{7\sqrt{10}} (3 	imes 2 + 0 	imes 3 + 1 	imes (-6)) = \frac{1}{7\sqrt{10}} (6 + 0 - 6) = 0$Since $\overline{a} \cdot \overline{b} = 0$,the vectors are perpendicular.Now,consider the expression $E = (2 \overline{a}-\overline{b}) \cdot [(\overline{a} 	imes \overline{b}) 	imes (\overline{a}+2 \overline{b})]$.Using the vector triple product identity $(\vec{u} 	imes \vec{v}) 	imes \vec{w} = (\vec{u} \cdot \vec{w})\vec{v} - (\vec{v} \cdot \vec{w})\vec{u}$:$(\overline{a} 	imes \overline{b}) 	imes (\overline{a}+2 \overline{b}) = (\overline{a} \cdot (\overline{a}+2 \overline{b}))\overline{b} - (\overline{b} \cdot (\overline{a}+2 \overline{b}))\overline{a}$$= (\overline{a} \cdot \overline{a} + 2(\overline{a} \cdot \overline{b}))\overline{b} - ((\overline{b} \cdot \overline{a}) + 2(\overline{b} \cdot \overline{b}))\overline{a}$$= (1 + 0)\overline{b} - (0 + 2(1))\overline{a} = \overline{b} - 2\overline{a}$Now substitute this back into the expression:$E = (2 \overline{a}-\overline{b}) \cdot (\overline{b} - 2\overline{a})$$= -(2 \overline{a}-\overline{b}) \cdot (2 \overline{a}-\overline{b}) = -|2 \overline{a}-\overline{b}|^2$$= -(4|\overline{a}|^2 + |\overline{b}|^2 - 4(\overline{a} \cdot \overline{b}))$$= -(4(1) + 1 - 4(0)) = -5$.

If $\overline{a}=\frac{1}{\sqrt{10}}(3 \hat{i}+\hat{k})$ and $\overline{b}=\frac{1}{7}(2 \hat{i}+3 \hat{j}-6 \hat{k})$,then the value of $(2 \overline{a}-\overline{b}) \cdot [(\overline{a} \times \overline{b}) \times (\overline{a}+2 \overline{b})]$ is

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