The direction ratios of normal to the plane through the points (0,-1,0) and (0,0,1) and ma…

Question

The direction ratios of normal to the plane through the points $(0,-1,0)$ and $(0,0,1)$ and making an angle $\dfrac{\pi}{4}$ with the plane $y-z+5=0$ are:

Accepted Answer

Let the equation of plane be

a(x - 0) + b(y + 1) + c(z - 0) = 0

It passes through (0, 0, 1) then

b + c = 0     . . . . (1)

Now cos $${\pi  \over 4}$$ = $${{a\left( 0 \right) + b\left( 1 \right) + c\left( { - 1} \right)} \over {\sqrt 2 \sqrt {{a^2} + {b^2} + {c^2}} }}$$

$$ \Rightarrow $$  a2 $=$ - 2bc  and  b $=$ - c

we get a2 $=$ 2c2

$$ \Rightarrow $$  a $=$ $$ \pm $$ $$\sqrt 2 $$ c

$$ \Rightarrow $$  direction ratio (a, b, c) = ($$\sqrt 2 $$, -1, 1)  or  ($$\sqrt 2 $$, 1, - 1)

Solutions · 1