General and Intercept Forms of Planes

Two Ways to Describe a Plane

A plane in three-dimensional space can be described by a linear equation in \(x\), \(y\), and \(z\). The general form displays the coefficients of those coordinates. The intercept form instead displays where the plane meets the three coordinate axes, when those intersections are suitable for this form.

These descriptions emphasize different features of the same plane. In the general form, the coefficients of \(x\), \(y\), and \(z\) are the components of a normal vector to the plane. In the intercept form, the denominators give the signed distances along the coordinate axes to the points where the plane meets those axes.

The condition that \(a,b,c\) are not all zero matters: if all three were zero, the equation would either be impossible or true at every point, rather than describing a plane. A nonzero scalar multiple of a plane equation describes the same plane. For instance, multiplying every term of a general equation by \(2\) does not change its solutions.

What the General-Form Coefficients Tell Us

In \(ax+by+cz+d=0\), the vector \(\mathbf{n}=(a,b,c)\) is a normal vector to the plane: it is perpendicular to directions that lie in the plane. This is consistent with the point-normal form of a plane equation. The constant \(d\) affects the plane's position, while \(a,b,c\) determine the direction of its normal. Multiplying all four coefficients by the same nonzero number changes the equation's appearance but not the plane or its normal direction.

The intercepts can be found directly from the general equation. To find the \(x\)-intercept, points on the \(x\)-axis have \(y=0\) and \(z=0\). If \(a\ne0\), substitution gives \(ax+d=0\), so \(x=-d/a\). The corresponding points on the other axes give the parallel formulas, provided the relevant coefficient is nonzero.

These are signed coordinate values, not necessarily positive lengths. For example, an \(x\)-intercept of \(-2\) means the plane meets the \(x\)-axis at \((-2,0,0)\). The sign is part of the equation and must be retained in intercept form.

Deriving Intercept Form

Suppose \(a,b,c\), and \(d\) are all nonzero. Starting from the general form, move the constant term to the right and divide by \(-d\). Then each term can be written using its axis intercept.

Since the intercepts are \(p=-d/a\), \(q=-d/b\), and \(r=-d/c\), their reciprocals are \(1/p=-a/d\), \(1/q=-b/d\), and \(1/r=-c/d\). These equal \(a/(-d)\), \(b/(-d)\), and \(c/(-d)\). Therefore the equation becomes \(\frac{x}{p}+\frac{y}{q}+\frac{z}{r}=1\). This derivation also shows exactly which conditions are needed: all three coefficients and the constant must be nonzero.

Worked Conversions

When Intercept Form Is Unavailable

There are two distinct reasons the three-intercept form may fail. First, a plane may meet one of the coordinate axes at no point. Second, it may pass through the origin, making its intercept on each axis that it meets equal to zero. Neither situation allows the equation \(\frac{x}{p}+\frac{y}{q}+\frac{z}{r}=1\) with three nonzero denominators.

If a coefficient such as \(c\) is zero, the \(z\)-coordinate does not appear in the general equation. Provided the equation still describes a plane, changing \(z\) does not affect whether a point satisfies it; the plane extends parallel to the \(z\)-axis. It may miss the \(z\)-axis entirely, or, if the other terms also vanish on that axis, contain the whole axis. In neither case is there a single \(z\)-intercept.

If \(d=0\), the origin satisfies the general equation. When \(a,b,c\) are all nonzero, setting each pair of coordinates to zero shows that all three axis intercepts are the origin. Dividing by those zero intercepts is impossible. For example, \(x+y+z=0\) contains the origin and meets each coordinate axis there, but no expression with three nonzero intercept denominators can represent those intercepts.

A Common Sign Error

A frequent mistake is to read the intercept as \(d/a\) instead of \(-d/a\). For example, from \(2x+3y-z-6=0\), an incorrect calculation might claim that the \(x\)-intercept is \(-3\). But on the \(x\)-axis, \(y=z=0\), so the equation is \(2x-6=0\), giving \(x=3\), not \(-3\). Substitution confirms the correction: at \((3,0,0)\), the left side is \(2(3)-6=0\); at \((-3,0,0)\), it is \(2(-3)-6=-12\), not zero.

Always find an intercept by setting the other two coordinates to zero in the original equation. This direct method makes the sign and any exceptional case visible.

Why Both Forms Are Useful

The general form is flexible: every plane has one, even when it passes through the origin or misses a coordinate axis. Its coefficients also display a normal vector immediately. Intercept form is more specialized, but it makes the three axis intersections easy to read. It can therefore be useful when a plane's intercepts are the information given or when the signed axis crossings are the feature of interest.

For a conversion, first test whether each axis intercept exists and is nonzero. If all three do, compute them by setting the other coordinates to zero and write the intercept form. If any coefficient or the constant is zero, do not force the equation into this form: retain the general equation and describe the missing or zero intercept accurately.