Algebra Examples

To find the intersection of the line through a point ${\displaystyle (p,q,r)}$ perpendicular to plane ${\displaystyle P_1}$ ${\displaystyle ax+by+cz=d}$ and plane ${\displaystyle P_2}$ ${\displaystyle ex+fy+gz=h}$:

1. Find the normal vectors of plane ${\displaystyle P_1}$ and plane ${\displaystyle P_2}$ where the normal vectors are ${\displaystyle n_1=〈a,b,c〉}$ and ${\displaystyle n_2=〈e,f,g〉}$. Check to see if the dot product is 0.

2. Create a set of parametric equations such that ${\displaystyle x=p+at}$, ${\displaystyle y=q+bt}$, and ${\displaystyle z=r+ct}$.

3. Substitute these equations into the equation for plane ${\displaystyle P_2}$ such that ${\displaystyle e(p+at)+f(q+bt)+g(r+ct)=h}$ and solve for ${\displaystyle t}$.

4. Using the value of ${\displaystyle t}$, solve the parametric equations ${\displaystyle x=p+at}$, ${\displaystyle y=q+bt}$, and ${\displaystyle z=r+ct}$ for ${\displaystyle t}$ to find the intersection ${\displaystyle (x,y,z)}$.

Next, build a set of parametric equations ${\displaystyle x=p+at}$,${\displaystyle y=q+bt}$, and ${\displaystyle z=r+ct}$ using the origin ${\displaystyle (0,0,0)}$ for the point ${\displaystyle (p,q,r)}$ and the values from the normal vector ${\displaystyle 5}$ for the values of ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$. This set of parametric equations represents the line through the origin that is perpendicular to ${\displaystyle P_1}$ ${\displaystyle x-y=4}$.

Substitute the expression for ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$ into the equation for ${\displaystyle P_2}$ ${\displaystyle 4x-y=-5}$.

Using the values calculated for ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$, the intersection point is found to be ${\displaystyle (-1,1,0)}$.