In a school worksheet, I have been given questions such as:
Given the line r : { 2x + y + z − 1= y + 2z = 0 }, find equations for the plane through r and passing through the point A=(2,1,0). and other questions that include finding planes parallel or orthogonal to a certain line or passing through certain points.
To solve these I've just been looking for solutions on the internet for each type of question since I can't find it in my university's textbook. It just includes a small chapter on cartesian and parametric equations.
Would anyone happen to know a good textbook or a website that explains this topic well?
Generally $ax+by+cz=d$ is a plane. Suppose you know a point on that plane $(x_0,y_0,z_0)$. Then $ax_0+by_0+cz_0=d.$ That means $a(x-x_0)+b(y-y_0)+c(z-z_0)=0$ by subtraction. In fact, if you know the normal and a point on the plane you can represent it with an equation of that form. Further, $<a,b,c>$ is the vector normal to the plane. A line is perpendicular to the plane if it is parallel to the normal.
Given three points on a plane $(x_1,y_1), (x_2,y_2), (x_3,y_3)$ we have:
$$a(x_1-x_2)+b(y_1-y_2)+c(z_1-z_2)=0$$ $$a(x_1-x_3)+b(y_1-y_3)+c(z_1-z_3)=0$$ $$a(x_2-x_3)+b(y_2-y_3)+c(z_2-z_3)=0$$
Those equations can be rearranged to solve for $a,b$ and $c$. That gives you the normal. As mentioned above, given a normal to a plane and a point on it, you can find the equation of that plane.
Given a line in a plane and a point not on that line, the equation of that plane can be found by taking two points on the line found by inspection (usually just let some combinations of coordinates be 0 and solve for those remaining), then using the point not on the line to find the normal via techniques mentioned above.
The shortest distance between a plane and a point not on that plane is
dist.$=\frac{a}{\sqrt{a^2+b^2+c^2}}(x_p-x_c)+\frac{b}{\sqrt{a^2+b^2+c^2}}(y_p-y_c)+\frac{c}{\sqrt{a^2+b^2+c^2}}(z_p-z_c)$
These are more efficiently expressed using vectors, but I'm assuming they aren't generally permitted.