Finding the Intercepts of a Plane

Intercept form

A plane's equation can be rewritten in intercept form, $\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1$ , where $a$ , $b$ , and $c$ are exactly where the plane crosses the $x$ -, $y$ -, and $z$ -axes.

Worked example

Find the sum of the intercepts cut off by the plane $2x+y-z=5$ .

Solution: Divide through by 5 to get the right-hand side to 1:

\dfrac{2x}{5}+\dfrac{y}{5}-\dfrac{z}{5}=1 \;\Rightarrow\; \dfrac{x}{5/2}+\dfrac{y}{5}+\dfrac{z}{-5}=1

So the intercepts are $a=\dfrac{5}{2}$ , $b=5$ , and $c=-5$ . Their sum:

\dfrac{5}{2}+5-5 = \dfrac{5}{2}

Finding the Intercepts of a Plane

Intercept form

A plane's equation can be rewritten in intercept form, $\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1$ , where $a$ , $b$ , and $c$ are exactly where the plane crosses the $x$ -, $y$ -, and $z$ -axes.

Worked example

Find the sum of the intercepts cut off by the plane $2x+y-z=5$ .

Solution: Divide through by 5 to get the right-hand side to 1:

\dfrac{2x}{5}+\dfrac{y}{5}-\dfrac{z}{5}=1 \;\Rightarrow\; \dfrac{x}{5/2}+\dfrac{y}{5}+\dfrac{z}{-5}=1

So the intercepts are $a=\dfrac{5}{2}$ , $b=5$ , and $c=-5$ . Their sum:

\dfrac{5}{2}+5-5 = \dfrac{5}{2}

Finding the Intercepts of a Plane

Intercept form

Worked example

Continue learning

Finding the Intercepts of a Plane

Intercept form

Worked example

Continue learning