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(B) The direction vectors of the two lines are $\vec{b_1} = \hat{\imath} - 2\hat{\jmath} - 2\hat{k}$ and $\vec{b_2} = 3\hat{\imath} + 2\hat{\jmath} -

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$\frac{11}{21}$

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(B) The direction vectors of the two lines are $\vec{b_1} = \hat{\imath} - 2\hat{\jmath} - 2\hat{k}$ and $\vec{b_2} = 3\hat{\imath} + 2\hat{\jmath} - 6\hat{k}$.The cosine of the angle $	heta$ between the lines is given by $\cos 	heta = \frac{|\vec{b_1} \cdot \vec{b_2}|}{|\vec{b_1}| |\vec{b_2}|}$.First,calculate the dot product: $\vec{b_1} \cdot \vec{b_2} = (1)(3) + (-2)(2) + (-2)(-6) = 3 - 4 + 12 = 11$.Next,calculate the magnitudes: $|\vec{b_1}| = \sqrt{1^2 + (-2)^2 + (-2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3$.$|\vec{b_2}| = \sqrt{3^2 + 2^2 + (-6)^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7$.Therefore,$\cos 	heta = \frac{|11|}{3 	imes 7} = \frac{11}{21}$.

The cosine of the angle included between the lines $\overline{r}=(2 \hat{\imath}+\hat{\jmath}-2 \hat{k})+\lambda(\hat{\imath}-2 \hat{\jmath}-2 \hat{k})$ and $\overline{r}=(\hat{\imath}+\hat{\jmath}+3 \hat{k})+\mu(3 \hat{\imath}+2 \hat{\jmath}-6 \hat{k})$ where $\lambda, \mu \in R$ is

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