Explain why there are many different vector and parametric equations for a plane.
Provide specific examples to justify your answer.
I answered an almost identical question here,
https://www.jiskha.com/questions/1828005/explain-why-there-are-many-different-vector-and-parametric-equations-for-a-plane-provide
Note that I pointed out a typo in my response which I corrected. It had no effect
on the explanation.
There are many different vector and parametric equations for a plane because there are multiple ways to represent its position and orientation in three-dimensional space. Each equation has its own advantages and is useful in different contexts.
To understand why there are multiple representations, let's start with the vector equation of a plane. A vector equation defines a plane by specifying a point on the plane and a normal vector perpendicular to the plane. The general form of the vector equation is:
r ⋅ n = r₀ ⋅ n,
where r is the position vector of any point on the plane, n is the normal vector of the plane, and r₀ is the position vector of a known point on the plane.
Now, let's consider some specific examples of vector equations for a plane:
1. Example 1: Standard Form
One common vector equation is in the standard form, which represents a plane using the coefficients of x, y, and z variables. The equation looks like this:
ax + by + cz = d,
where a, b, c are the coefficients of the normal vector, and d is a constant. This form is useful for understanding the equation of a plane in terms of its coefficients.
2. Example 2: Intercept Form
The intercept form of the vector equation represents a plane using the distances from the origin to the intersections with the three coordinate axes. The equation is given as:
x/a + y/b + z/c = 1,
where a, b, c are the reciprocals of the intercepts on the x, y, and z axes, respectively. This form is helpful when you need to determine the intercepts of the plane on the coordinate axes.
Now, let's discuss parametric equations for a plane. A parametric equation describes a plane by specifying how the coordinates of points on the plane vary with two independent parameters (usually denoted by u and v). Parametric equations are useful when you want to describe the plane as a combination of two varying parameters.
Here's an example of a parametric equation for a plane:
3. Example 3: Parametric Form
The parametric form of a plane equation is given by:
x = a + su + tv,
y = b + su' + tv',
z = c + su'' + tv'',
where a, b, and c define a point on the plane, and s and t are the parameters. The vectors u and v are usually chosen as two non-parallel vectors lying in the plane. This form of equation is useful when you want to describe the motion or variation of points on the plane in terms of two independent parameters.
In summary, the reason there are many different vector and parametric equations for a plane is that each representation provides different insights and serves different purposes. The standard form gives you the coefficients of the normal vector, the intercept form provides information about the intercepts with the coordinate axes, and the parametric form allows for describing the plane as a combination of two parameters.