Find the coordinates of the point, where the line (x-2)/3=(y+1)/4=(z-2)/2 intersects the plane x − y + z − 5 = 0. Also find the angle between the line and the plane. - Mathematics

Find the coordinates of the point, where the line (x-2)/3=(y+1)/4=(z-2)/2 intersects the plane x − y + z − 5 = 0. Also find the angle between the line and the plane.

Solution

The equation of the given line is (x-2)/3=(y+1)/4=(z-2)/2 ...(1)

Any point on the given line is (3lambda+2,4lambda-1,2lambda+2)

If this point lies on the given plane xy + z − 5 = 0, then 3lambda+2-(4lambda-1)+2lambda+2-5=0

lambda=0

Putting lambda =0 in (3lambda+2,4lambda-1,2lambda+2),we get (3lambda+2,4lambda-1,2lambda+2)=(2,-1,2)

So, the point of intersection of the given line and the plane is (2,-1,2)

Let θ be the angle between the given line and the plane.

therefore sin theta=(veca.vecb)/(|veca|.|vecb|)=((3hati+4hatj+2hatk)(hati-hatj+hatk))/(sqrt(3^2+4^2+2^2)sqrt(1^2_(-1)^2+1^2))

=(3xx14xx(-1)+2xx1)/(sqrt(29)sqrt(3))=1/sqrt87

theta=sin^(-1)(1/sqrt87)

Thus, the angle between the given line and the plane is sin ^(-1)(1/sqrt87).

Concept: Angle Between Line and a Plane
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