Problem 26 - Calculus an Applied Approach

Question

Determine whether the planes \(a_{1} x+b_{1} y+c_{1} z=d_{1}\) and \(a_{2} x+b_{2} y+c_{2} z=d_{2}\) are parallel, perpendicular, or neither. The planes are parallel if there exists a nonzero constant \(k\) such that \(a_{1}=k a_{2}, b_{1}=k b_{2}\), and \(c_{1}=k c_{2},\) and are perpendicular if \(a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0\). $$ x+3 y+z=7, x-5 z=0 $$

Step-by-Step Solution

Answer

The given planes are neither parallel nor perpendicular.

Step 1: Identify the coefficients of the equations

The coefficients \(a_1, b_1, c_1\) for the first plane are 1, 3, and 1 respectively. Similarly, the coefficients \(a_2, b_2, c_2\) for the second plane are 1, 0, and -5.

Step 2: Check if the planes are parallel

Two planes are parallel if there exists a non-zero constant \(k\) such that \(a_1 = ka_2, b_1 = kb_2\), and \(c_1 = kc_2\). Here, no such constant k exists that satisfies the condition, so the planes are not parallel.

Step 3: Check if the planes are perpendicular

Planes are perpendicular if \(a_1a_2 + b_1b_2 + c_1c_2 = 0\). Substituting the values, we get \(1*1 + 3*0 + 1*(-5) = 1 - 5 = -4\), which is not equal to zero. Therefore, the planes are not perpendicular.

Step 4: Conclusion

Because the planes are not parallel or perpendicular, the relationship between the two planes is classified as neither.

Key Concepts

Parallel Planes

Parallel planes in geometry are like two sheets of paper stacked on a shelf, never touching, no matter how far they extend. They have the same orientation, running in the same direction in space. To determine if two planes are parallel, we compare their normal vectors.

Each plane can be expressed in the general form: \(a_1x + b_1y + c_1z = d_1\) and \(a_2x + b_2y + c_2z = d_2\). Since the normal vector of each plane is defined by its coefficients \(a_1, b_1, c_1\) and \(a_2, b_2, c_2\), two planes are parallel if there is a constant \(k\) such that:

• \(a_1 = ka_2\)
• \(b_1 = kb_2\)
• \(c_1 = kc_2\)

For the planes in our exercise, \(k\) does not exist as no single constant can equate all the corresponding coefficients simultaneously, confirming that they are not parallel.

In simpler terms, if you can't find a common factor for the coefficients of the planes' equations, the planes aren't parallel.

Perpendicular Planes

Perpendicular planes intersect at right angles (90 degrees). Imagine the corner of a room, where the wall meets the floor. This is what perpendicular planes would look like; they meet forming a 'T' shape.

To determine if two planes are perpendicular, we use the dot product of their normal vectors. If the dot product of the normal vectors is zero, then the planes are perpendicular.

\(a_1a_2 + b_1b_2 + c_1c_2 = 0\) needs to hold true. This means the work done by one vector when projected onto the other is zero, indicating they are orthogonal.

In the provided equations, the calculation gives \(1*1 + 3*0 + 1*(-5) = -4\), which is not zero. Thus, these planes are not perpendicular because their intersection isn’t a right angle.

Equation of a Plane

The equation of a plane is a fundamental concept in geometry that describes a flat, two-dimensional surface extending infinitely in three-dimensional space. The general form for the equation of a plane is \(ax + by + cz = d\). Here, \(a, b,\) and \(c\) are the coefficients representing the normal vector to the plane. The normal vector is essential because it defines the plane’s orientation in space.

Consider \(x + 3y + z = 7\) and \(x - 5z = 0\).

• The first equation can also be perceived as altering the line equation by shifting it perpendicularly to \(x\)-axis, \(y\)-axis, and \(z\)-axis.
• The second equation lacks a term for \(y\), meaning it’s aligned parallel to the \(y\)-axis.

These equations show different orientations and positions in a space grid, allowing us to understand how these planes may interact with each other, given their described spatial positions.