本题目来源于试卷: VectorsVectors，类别为 IB数学
[问答题]
Two planes have equatis(g8w/t l lkz67j/a cueh5s lu yc(+ub(w.4v nons
$\Pi_{1}$: 2 x-2 y+z=1 $\text { and } \Pi_{2}: x-4 y-z=5 $.
1. Find the cosine of the angle between the two planes, giving your answer in the form $\frac{\sqrt{p}}{q}$ where p, q$\in \mathbb{Z}^{+} $.
Let L be the line of intersection of the two planes.
2. 1. Show that L has direction $ 2 \mathbf{i}+\mathbf{j}-2 \mathbf{k}$.
2. Show that the point A(4,1,-5) lies on both planes.
3. Write down the vector equation of L .
B is the point on $ \Pi_{1}$ with coordinates (a, b, 3) .
3. Given that the vector $\overrightarrow{\mathrm{AB}}$ is perpendicular to L , find the value of a and the value of b .
4. Show that $ \mathrm{AB}=12$ .
The point C lies on L and $\mathrm{AB} \mathrm{B}=60^{\circ}$ .
5. Find the coordinates of the two possible positions of C .
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