VectorsVectors (id: 9214f5896)

本题目来源于试卷: VectorsVectors，类别为 IB数学

[问答题]
Two planes have equap1)od ka6n c3wfx *(kf5eapc- w)xt,otions

${\mathrm{\Pi }}_1$: 2 x+4 y+z=9 $\text{ and }{\mathrm{\Pi }}_2$: 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 ${\mathrm{\Pi }}_1$ 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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