Problem

The line $l_1$ has equation

$$\frac{x-3}{4}=\frac{y-5}{-2}=\frac{z-4}{7}$$

The plane $\Pi$ has equation

$$2x+4y-z=1$$

The line $l_1$ intersects the plane $\Pi$ at the point $P$

(a) Determine the coordinates of $P$

The acute angle between $l_1$ and $\Pi$ is $\theta$ degrees.

(b) Determine, to one decimal place, the value of $\theta$

The line $l_2$ lies in $\Pi$ and passes through $P$

Given that the acute angle between $l_1$ and $l_2$ is also $\theta$ degrees,

(c) determine a vector equation for $l_2$