Find a plane containing the line x(t)=−2+6t, y(t)= 7−7t, z(t)=3−t and orthogonal to the plane −5x−4y+8z=4

Question

Find a plane containing the line 

x(t)=−2+6t

y(t)= 7−7t

z(t)=3−t 

and orthogonal to the plane −5x−4y+8z=4

Answer 1

We take 2 points on this line: t = 0 gives point (- 2, 7, 3) and t = 1 gives point (4, 0, 2).

Equation of plane: ax + by + cz + d = 0

-2a + 7b + 3c + d = 0

4a + 2c + d = 0 and vector <a, b, c> orthogonal to vector <-5, - 4, 8>.

So, -5a - 4b + 8c = 0.

You have to solve thissystem to express a, b and c in terms of d and then put in equation of plane and reduce byd