本题目来源于试卷: VectorsVectors，类别为 IB数学
[问答题]
Two planes have equationnys95s8yp 5pu su4nyd 3 u20itu
$\Pi_{1}$: 2 x+4 y+z=9 $\text { and } \Pi_{2}$: 2 x+y-z=1 .
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 $5 \mathbf{i}-4 \mathbf{j}+6 \mathbf{k}$ .
2. Show that the point P(0,2,1) lies on both planes.
3. Write down the vector equation of L .
Q is the point on $\Pi_{1} $ with coordinates (a, 1, b) .
3. Given that the vector $\overrightarrow{\mathrm{PQ}} $ is perpendicular to L , find the value of a and the value of b .
4. Show that $\mathrm{PQ}=\sqrt{33}$ .
The point R lies on L and $\mathrm{PQ} \mathrm{Q}=30^{\circ} $.
5. Find the coordinates of the two possible positions of R.
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