The direction ratios of the normal to the plane through (1,0,0), (0,1,0), which makes an a…

Question

The direction ratios of the normal to the plane through $(1,0,0)$, $(0,1,0)$, which makes an angle $\pi/4$ with the plane $x+y=3$, are

Accepted Answer

Plane $a(x-1)+by+cz=0$ through $(1,0,0)$; through $(0,1,0)$: $-a+b=0\Rightarrow b=a$.
$$\cos45^{\circ}=\frac{|a+a|}{\sqrt2\sqrt{2a^2+c^2}}\implies 2a=\sqrt2\sqrt{2a^2+c^2}\implies 2a^2=c^2\implies c=\sqrt2\,a.$$
Direction ratios: $a,a,\sqrt2a$, i.e. $1,1,\sqrt2$.

Solutions · 1