Problem 32 - Algebra and Trigonometry

Question

A description of a plane is given. Find an equation for the plane. The plane that crosses the \(x\) -axis where \(x=-2,\) the \(y\) -axis where \(y=-1,\) and the \(z\) -axis where \(z=3\)

Step-by-Step Solution

Answer

The plane equation is \(-3x - 6y + 2z = 6\).

Step 1: Understand the Intersection Points

The plane intersects the x-axis at \((-2, 0, 0)\), the y-axis at \((0, -1, 0)\), and the z-axis at \((0, 0, 3)\). These are the taken points of intersection in the respective axes.

Step 2: Use the Intercept Form of a Plane Equation

The general intercept form of a plane equation is \[\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\] where \(a, b, c\) are the intercepts on the x, y, and z axes, respectively. In this case, \(a = -2\), \(b = -1\), and \(c = 3\).

Step 3: Substitute Intercept Values Into the Equation

Substitute the intercepts into the intercept form equation: \[\frac{x}{-2} + \frac{y}{-1} + \frac{z}{3} = 1\]

Step 4: Simplify the Equation

To simplify, clear the fractions by multiplying through by the least common multiple (LCM) of the denominators. Here, the LCM is 6.The equation becomes:\[-3x - 6y + 2z = 6\].

Step 5: Finalize the Plane Equation

The simplified equation \(-3x - 6y + 2z = 6\) is the equation of the plane in standard form, where coefficients of \(x, y, z\) suggest the direction of the normal vector.

Key Concepts

Intercepts on Axes

When a plane intersects the three-dimensional axes, it creates what are known as intercepts. An intercept is simply the point at which a plane crosses an axis, and they are crucial in defining the plane's equation. In this exercise, we have been given three intercepts: the plane crosses the x-axis where \( x = -2 \), the y-axis where \( y = -1 \), and the z-axis where \( z = 3 \). These intercepts are represented as the coordinates \((-2, 0, 0)\), \((0, -1, 0)\), and \((0, 0, 3)\), respectively.

Each intercept provides essential information because it represents how the plane "cuts" through space at these pivotal points. Understanding intercepts helps in obtaining a clear picture of the plane's position in space.

Intercept Form of Plane

The intercept form of the plane's equation is a straightforward way to define a plane using its axis crossings. The general equation is written as:

• \(\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\)

where \(a, b,\) and \(c\) are the intercepts on the x, y, and z axes, respectively. This form is particularly useful because it directly incorporates these intercept points, making it simple to plug in the given values.

By substituting our known intercepts \(a = -2\), \(b = -1\), and \(c = 3\) into this equation, we get:

• \(\frac{x}{-2} + \frac{y}{-1} + \frac{z}{3} = 1\)

This intercept form highlights the proportional relationship between the coordinates, emphasizing how each axis contributes to forming the plane. Transforming this into standard form often involves simplifying by removing fractions to express it in a more familiar linear manner.

Normal Vector

The normal vector is a key concept in understanding the orientation of a plane in 3D space. It is a vector that is perpendicular to the plane, pointing in a distinct direction. In the standard form equation of the plane, which we derived as \(-3x - 6y + 2z = 6\), the coefficients of \(x, y,\) and \(z\) (i.e., \(-3\), \(-6\), and \(2\)) represent the components of the plane's normal vector.

Thus, the normal vector \(\mathbf{n}\) can be expressed as:

• \(\mathbf{n} = \langle -3, -6, 2 \rangle\)

Knowing the normal vector is crucial for several applications including calculating the angle between two planes or determining a plane's slant in relation to the coordinate axes. Its perpendicular nature also allows it to define the plane's orientation uniquely and efficiently, providing insight into geometrical problems where orientation is important.