Equation of a plane in intercept form
Equation of a Plane in Intercept Form
In three-dimensional geometry, a plane can be defined in various forms, each providing different insights or computational advantages. One such form is the intercept form, which is particularly useful when a plane intersects the coordinate axes at non-zero points.
Understanding the Intercept Form
The intercept form of the equation of a plane is based on the x, y, and z-intercepts of the plane. These intercepts are the points where the plane crosses the x, y, and z-axes, respectively.
If a plane intersects the x-axis at (a), the y-axis at (b), and the z-axis at (c), then the intercept form of the equation of the plane is given by:
[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 ]
where (a), (b), and (c) are the x, y, and z-intercepts of the plane, respectively.
Important Points and Differences
Here's a table summarizing the key points and differences between the intercept form and other forms of the equation of a plane:
| Feature | Intercept Form | General Form | Point-Normal Form |
|---|---|---|---|
| Equation | (\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1) | (Ax + By + Cz + D = 0) | ((\vec{r} - \vec{r_0}) \cdot \vec{n} = 0) |
| Intercepts | Directly given by (a), (b), and (c) | Not directly available | Not directly available |
| Normal Vector | Not directly available | The coefficients (A), (B), and (C) | The vector (\vec{n}) |
| Point on Plane | ((a, 0, 0)), ((0, b, 0)), ((0, 0, c)) | Requires calculation | The point (\vec{r_0}) |
| Applicability | Only for planes that intersect all three axes | For any plane | For any plane with a known normal and point |
Formulas and Examples
Example 1: Finding the Equation of a Plane
Suppose a plane intersects the x-axis at (4), the y-axis at (3), and the z-axis at (2). The intercept form of the equation of this plane is:
[ \frac{x}{4} + \frac{y}{3} + \frac{z}{2} = 1 ]
Example 2: Converting to General Form
To convert the intercept form to the general form, we can multiply through by the denominators:
[ \frac{x}{4} + \frac{y}{3} + \frac{z}{2} = 1 \implies 3 \cdot 2 \cdot x + 4 \cdot 2 \cdot y + 4 \cdot 3 \cdot z = 4 \cdot 3 \cdot 2 ]
Simplifying, we get:
[ 6x + 8y + 12z = 24 \implies 6x + 8y + 12z - 24 = 0 ]
Example 3: Finding Intercepts from General Form
Given the general form of the equation of a plane:
[ 2x + 3y + 6z - 12 = 0 ]
To find the intercepts, we set two variables to zero and solve for the third:
- For the x-intercept, set (y = 0) and (z = 0):
[ 2x = 12 \implies x = 6 ]
- For the y-intercept, set (x = 0) and (z = 0):
[ 3y = 12 \implies y = 4 ]
- For the z-intercept, set (x = 0) and (y = 0):
[ 6z = 12 \implies z = 2 ]
Now we can write the equation in intercept form:
[ \frac{x}{6} + \frac{y}{4} + \frac{z}{2} = 1 ]
Example 4: Finding a Point on the Plane
Given the intercept form of a plane:
[ \frac{x}{5} + \frac{y}{-7} + \frac{z}{3} = 1 ]
We can find a point on the plane by setting two variables to zero and solving for the third:
- A point on the x-axis is ((5, 0, 0))
- A point on the y-axis is ((0, -7, 0))
- A point on the z-axis is ((0, 0, 3))
Conclusion
The intercept form of the equation of a plane is a convenient way to represent a plane when its intercepts with the coordinate axes are known. It provides a direct way to visualize the plane's position in 3D space and can be easily converted to other forms for further analysis or computation. Understanding this form is essential for solving problems in 3D geometry, especially in the context of exams where quick and efficient methods are required.