Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

The equation of the plane is $\vec{r}\cdot (3\hat{i}-2\hat{j}+7\hat{k})=32$.

The normal vector to the plane is $\vec{n}=3\hat{i}-2\hat{j}+7\hat{k}$.

The direction vector of the line is $\vec{d}=5\hat{i}+b\hat{j}+3\hat{k}$.

Since the line is parallel to the plane, its direction vector is perpendicular to the normal vector of the plane.

$\vec{n}\cdot \vec{d}=0$

$(3)(5)+(-2)(b)+(7)(3)=0$

$15-2b+21=0$

$36-2b=0$

$b=18$

Answer: $b=18$