Find the scalar equation of the plane that is perpendicular to the plane x+2y+4=0, contains the origin, and whose normal makes an angle of 30 with the z-axis.
attempt:
d1= (1,2,0)
P(0,0,0)
cos(theta)=z/|normal|
As I assume Soroban tried to attempt this problem algebraically. I've choosen a completely different way to do this question:
1. From the equation of the plane $\displaystyle p_1$ you know that the normal vector is $\displaystyle \overrightarrow{n_1} = (1,2,0)$ . That means $\displaystyle p_1$ is perpendicular to the x-y-plane.
2. I looked for a line which lies simultaneously in $\displaystyle p_1$ and the x-y-plane and took the line which connects the intersection points of $\displaystyle p_1$ and the x-axis and $\displaystyle p_1$ and the y-axis. The intersection points are A(-4,0,0) and B(0,-2,0).
3. I took the line $\displaystyle \vec r= t \cdot (1,2,0)$ as a kind of axis. I then "tilted" the x-y-plane (which is perpendicular to $\displaystyle p_1$!) about this "axis" by 30° to get $\displaystyle p_2$. If I tilt the x-y-plane by 30° then the normal vector of $\displaystyle p_2$ and the z-axis include an angle of 30°.
4. I took the vector $\displaystyle \overrightarrow{AB} =(-4,2,0)$ and rotated it by 30° to get the vector $\displaystyle \vec b = \left(-4,2,\sqrt{\frac{20}3}\right)$ (By the way: You probably have noticed that there must exist a second vector $\displaystyle \vec b '$ if you rotate the x-y-plane in the opposite direction)
5. Now I know 2 vectors which span the plane $\displaystyle p_2$. Since $\displaystyle p_2$ has to contain the origin the parametric equation of $\displaystyle p_2$ is:
$\displaystyle p_2$: $\displaystyle \vec r = s \cdot (1,2,0) + t \cdot \left(-4,2,\sqrt{\frac{20}3}\right)$ ....... or ....... $\displaystyle p_2: \left\{\begin{array}{lcr}x&=&s-4t \\ y&=&2s+2t \\ z&=&\sqrt{\frac{20}3} \cdot t \end{array}\right.$
6. The cross product of the 2 direction vectors which span $\displaystyle p_2$ is the normal vector of $\displaystyle p_2$:
$\displaystyle (1,2,0) \times \left(-4,2,\sqrt{\frac{20}3}\right) = \left(\frac43 \cdot \sqrt{15}\ ,\ -\frac23 \cdot \sqrt{15}\ ,\ 10 \right)$
And therefore the scalar equation of $\displaystyle p_2$ is:
$\displaystyle \boxed{\frac43 \cdot \sqrt{15} \cdot x - \frac23 \cdot \sqrt{15} \cdot y + 10 \cdot z = 0}$
7. I finally used derive 6.0 to draw the 2 planes and the normal vector. This program is able to rotate the 3-D-graphs about an axis. I took 2 screenshots to show how the planes and the normal are placed.