Calculate the intercepts on the coordinate axes cut by the…
Calculate the intercepts on the coordinate axes cut by the planes: a) $x-2y+3z-6=0$; b) $5x+y-3z-15=0$; c) $x+2y-3z=0$; d) $2y-3z-6=0$; e) $x-3=0$.
Explanation of Method
The intercepts cut on the coordinate axes by five planes.
Step 1: Method
Set two coordinates to zero and solve for the third. If a coordinate is missing from the equation, the plane is parallel to that axis and has no intercept on it.
Step 2: Results
| Plane | $a$ | $b$ | $c$ | Intercept form | |
|---|---|---|---|---|---|
| a) | $x-2y+3z=6$ | $6$ | $-3$ | $2$ | $\tfrac x6+\tfrac y{-3}+\tfrac z2=1$ |
| b) | $5x+y-3z=15$ | $3$ | $15$ | $-5$ | $\tfrac x3+\tfrac y{15}+\tfrac z{-5}=1$ |
| c) | $x+2y-3z=0$ | $0$ | $0$ | $0$ | none (through $O$) |
| d) | $2y-3z=6$ | none ($\parallel Ox$) | $3$ | $-2$ | $\tfrac y3+\tfrac z{-2}=1$ |
| e) | $x=3$ | $3$ | none | none | $\tfrac x3=1$ ($\parallel yOz$) |
Result
\[\begin{aligned} a)&\ a=6,\ b=-3,\ c=2\\ b)&\ a=3,\ b=15,\ c=-5\\ c)&\ a=b=c=0\ \text{(through }O\text{)}\\ d)&\ \text{no }a\ (\parallel Ox),\ b=3,\ c=-2\\ e)&\ a=3,\ \text{no }b,\ \text{no }c\ (\parallel yOz) \end{aligned}\]
Remark
The source writes “$a=\infty$” for a missing intercept. It is clearer to say the plane is parallel to that axis and has no intercept on it. (Formally, $\tfrac{x}{a}\to0$ as $a\to\infty$.)