Class 12

Math

3D Geometry

Three Dimensional Geometry

Find the angle between the line $2x+1 =3y =6z−3 $ and the plane $10x+2y−11z=3$.

Direction ratios of the given line are $2,3,6$

Direction ratios of the normal to the given plane are $10,2,−11$

Now, the angle between the line and the plane is given by:

$sinθ=∣a∣∣n∣∣a.n∣ $

$sinθ=a_{1}+b_{1}+c_{1} .a_{2}+b_{2}+c_{2} ∣a_{1}.a_{2}+b_{1}.b_{2}+c_{1}.c_{2}∣ $

$∴sinθ={2_{2}+3_{2}+6_{2} }{(10)_{2}+2_{2}+(−11)_{2} }∣(2×10)+(3×2)+6×(−11)∣ $

$={49 }×{225 }40 =(7×15)40 =218 $

$⇒θ=sin_{−1}(218 )$.