Problems from Angle between two straight lines

Determine the angle between $r$ and $s$ :

$r : \frac{x + 1}{3} = \frac{y - 2}{- 1} = \frac{z + 1}{3} s : {\begin{cases} x = 1 - k \\ y = 2 + k \\ z = k \end{cases}$

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Development:

We start by looking for governing vectors of $r$ and $s$ :

a governing vector of $r$ is $\vec{u} = (3, - 1, 3)$ .
a governing vector of $s$ is $\vec{v} = (- 1, 1, 1)$ .

We can already apply the formula:

$\begin{aligned} \cos (α) = & \frac{| u_{1} v_{1} + u_{2} v_{2} + u_{3} v_{3} |}{\sqrt{u_{1}^{2} + u_{2}^{2} + u_{3}^{2}} \sqrt{v_{1}^{2} + v_{2}^{2} + v_{3}^{2}}} = \frac{| 3 \cdot (- 1) + (- 1) \cdot 1 + 3 \cdot 1 |}{\sqrt{3^{2} + (- 1)^{2} + 3^{2}} \sqrt{(- 1)^{2} + 1^{2} + 1^{2}}} \\ = & \frac{1}{\sqrt{19} \sqrt{3}} = 0.13245 \end{aligned}$

Therefore,

$α = \arccos (0.13245) = {82.39}^{\circ}$

Solution:

$α = {82.39}^{\circ}$

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