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I am having problems getting the answer for (ii) Skew, $\angle$ between lines $= 48.5^{\circ}$ I am getting 144. Perhaps I did something wrong? Actually what does the dot product ($a \cdot b = x_1x_2+y_1y_2+z_1z_2$) give?

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The dot product gives $\mathbf{a} \cdot \mathbf{b}=\left\|\mathbf{a}\right\| \, \left\|\mathbf{b}\right\| \cos \theta$

You have an error in what you call $DirVec_1$: you have read $3\mathbf{i}+\mathbf{j}-3\mathbf{k}$ as having coefficients $3,0,-3$ when it should be $3,1,-3$.

Do your calculations again and you will get $\cos^{-1}\left(\dfrac{-5}{\sqrt{19}\sqrt{3}}\right) \approx 131.47^{\circ}$. Since a line makes two angles with another line, you can also subtract this from $180^{\circ}$ to get the answer you expect.

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