Planes ${{p}_{1}}$ and ${{p}_{2}}$ are respectively represented by cartesian equations $x+2y+2z=3,$

Question

Planes ${{p}_{1}}$ and ${{p}_{2}}$ are respectively represented by cartesian equations $x+2y+2z=3,$ $-2x-y+2z=-6,$ and they meet in a line $l$. Find a vector equation for line $l$ . [2] Plane ${{p}_{3}}$ is represented by cartesian equation $M\left( x+2y+2z-3 \right)+\left( -2x-y+2z+6 \right)=0$, where $M$ is a constant. Write down a normal vector to plane ${{p}_{3}}$, in terms of $M$. [1] Explain why plane ${{p}_{3}}$ contains line $l$, for any value of $M$ . [2] Show that the cosine of the acute angle between planes ${{p}_{1}}$ and ${{p}_{3}}$ is $\frac{M}{\sqrt{{{M}^{2}}+1}}$ , if $M>0$. [4]

Accepted Answer

l: r = (3, 0, 0) + λ(2, -2, 1); n₃ = (M-2, 2M-1, 2M+7); P₃ contains l; cos θ = M/√(M²+1)

2019 CJC Promo Q9

2019 CJC Promo Q9

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