A plane is a flat two dimensional surface in three dimensional space. You can hold up a book or an ipad in front of you to visualise a plane, and your plane can be rotated and translated in space to view unique planes. The Cartesian equation of a plane has the form
\( ax+by+cz=d \)
where \(a \), \(b \), \(c \) and \(d\) are constants. The Cartesian form has been stated without proof or motivation so far, for now it's useful to analyse the Cartesian equation to visualize planes in 3D space.
Consider the plane \( ax+by+cz=d \). All points on the x axis have \(y=0\) coordinate and \(z=0\). Therefore, the intersection point \(A\) of the plane with the x axis can be found by setting \(y=0\) and \(z=0\) and then solving for \(x \) :
\(ax=d \) \( \Rightarrow \) x=q and \( A\left (0,0,0 \right ) \).
Similarly, by setting \(x=0\) and \(z=0\) and solving for \(y\), to find the \(y\) intersect coordinate \( B\left (0,0,0 \right ) \).
then by setting \(x=0\) and \(y=0\) and solving for z the \(z\) intersect coordinate \( C\left (0,0,0 \right ) \).
Write down the equation of the plane intersects the x axis for x=2, the y axis for y=-1, and the z axis fro z=3
Click to view the answer
In two dimensions vertical and horizontal lines are not explicitly written in the form y=mx+c, for example x=2 is a vertical line and y=3 is a horizontal line. In three dimension the plane x=2 would be a plane that is parallel to the y-z plane where all points have x cooordinate x=2.
\( \mathbf{a}= \begin{bmatrix} 1 \\ 3 \end{bmatrix} \)
Use the sliders to change the values of a,b,c, and d. The distance of the plane from the origin and the intersects with axes and the planes orientation can be changed.