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I have this problem, I have to find the values of $a$ so the direction vectors of the lines make an angle of $60$ degrees.

$$\frac{x-3}{2} = \frac{y+5}{2} = z+2$$ $$ x-1 = y-1 = \frac{z-3}{a}$$

What should be the logic to develop this exercise?

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You could:

1) Find the direction vectors $\bf v$ and $\bf w$ of the two lines.

2) Solve ${\bf v}\cdot {\bf w}=\Vert {\bf v}\Vert \Vert {\bf w}\Vert \cos 60^\circ$.

The direction vectors are ${\bf v}=(2,2,1)$ for the first line and ${\bf w}=(1,1,a)$ for the second.
We have $$\eqalign{ {\bf v}\cdot {\bf w} &=2\cdot1+2\cdot1+1\cdot a\cr &= 4+a}$$ and $$ \Vert {\bf v}\Vert =\sqrt9=3,\qquad \Vert {\bf w}\Vert=\sqrt{2+a^2}. $$ So, you have to solve $$ 4+a= 3\cdot\sqrt{2+a^2}\cdot\textstyle{1\over2}.$$

If you square both sides of the above, you'll have a quadratic equation in $a$ with real solutions.

I believe (but haven't checked) that both solutions will "work".

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