Question

Find the equation of a plane passing through the points A(2,1,2) and B(4, -2,1) and perpendicular to plane vector r.(i - 2k) = 5. Also, find the coordinates of the point, where the line passing through the points (3,4,1) and (5,1,6) crosses the plane thus obtained.

Answer 1

Let P(x,y,z) be any point on the plane in which A (2,1,2) and B (4, -2, 1) lie.

∴ vector (AP and AB) lie on required plane.

Also required plane is perpendicular to given plane vector r.(i - 2k) = 5

∴ normal to given plane vector n₁ = (i - 2k) lie on required plane.

vector (AP, AB and n₁) are coplanar.

where vector AP = (x - 2)i + (y - 1)j + (z - 2)k

vector AB = 2i - 3j - k

Scaler triple product vector [AP AB n₁] = 0

Line passing through points L(3,4,1) and M(5,1,6) is 

⇒ General point on the line is Q(2λ + 3), (-3λ + 4), (5λ + 1)

As line (2) crosses plane (1) so point Q should satisfy equation(1)