Equation of Plane with Line and Angle Problem

[chmate]

Homework Statement

Find the equation of plane which contains the line l:\left\{\begin{array}{l} x=t+2 \\y=2t-1\\z=3t+3 \end{array}\right., and makes the angle of \frac{2\pi}{3} with the plane \pi:x+3y-z+8=0.

The Attempt at a Solution

My attempt was to find the normal vector of plane which contains the line by the cross product of vector \vec{a} of line and some vector created between points P(2,-1,3) of line and point Q(x,y,z), than by using the formula to find angles, but this leads me to complications.

I think there's another way to solve this, which I don't know.

Thank you

 

[HallsofIvy]

If two planes make a given angle, \theta, with each other, then their normal vectors make the same angle. The dot product of two vectors, u and v, is given by u\cdot v= |u||v|cos(\theta). So if we take <a, b, c> to be a unit vector perpendicular to the desired plane, we must have &lt;a, b, c&gt;\cdot&lt;1, 3, -1&gt;= \sqrt{1+ 9+ 1} cos(2\pi/3) or a+ 3b- c= -.5\sqrt{11}. That, together with a^2+ b^2+ c^2= 1 gives two equations to solve for a, b, and c.