Problem 25 - 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 \(a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0\). $$ x+2 y=3,4 x+8 y=5 $$

Step-by-Step Solution

Answer

The given planes are parallel.

Step 1: Identify the Plane Equations

From the exercise, the two given planes are represented by the equations \(x + 2y = 3\) and \(4x + 8y = 5\).

Step 2: Identify the Coefficients of the Planes

For the first equation, \(a1=1\), \(b1=2\) and \(c1=0\) (since there's no z term). For the second equation, \(a2=4\), \(b2=8\) and \(c2=0\). There is no z term in either plane equation, thus their respective c coefficients are zero.

Step 3: Determine if Planes are Parallel

Planes are parallel if \(a1 = k * a2\), \(b1 = k * b2\) and \(c1 = k * c2\) for some constant k. Let's set k as the ratio of \(a1\) to \(a2\), so \(k = a1 / a2 = 1 / 4\). Now check if the same k also holds for \(b1\) and \(b2\). \(b1 / b2 = 2 / 8 = 1 / 4 = k\). The c coefficients are both zero so \(c1 = c2\). Since k equals to the ratio of all corresponding coefficients of the planes, the planes are therefore parallel.

Step 4: Checking Perpendicularity (Optional)

Since the planes were found to be parallel in Step 3, it's not necessary to check for perpendicularity as they can't both be. However, for completeness, planes will be perpendicular if \(a1*a2 + b1*b2 + c1*c2 = 0\). \(a1*a2 + b1*b2 + c1*c2 = 1*4 + 2*8 + 0*0 = 20 ≠ 0\). The planes are not perpendicular.

Key Concepts

Parallel Planes

In plane geometry, understanding when two planes are parallel is fundamental. Parallel planes have the same orientation and will never intersect, no matter how far they expand. For two planes to be parallel, the ratios of their respective coefficients,

• If the first plane is defined as \(a_1x + b_1y + c_1z = d_1\) and the second as \(a_2x + b_2y + c_2z = d_2\), then:
• \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)

In the example given, we identified that the planes \(x + 2y = 3\) and \(4x + 8y = 5\) are parallel. Setting \(k = \frac{1}{4}\), which is derived from \(\frac{a_1}{a_2}\), confirms that the constant relation holds across the coefficients \(b_1/b_2\) and that \(c_1 = c_2\) (both being zero). Hence, this implies that the planes never meet. A consistent ratio across coefficients is the clue to parallelism.

Perpendicular Planes

When tackling perpendicular planes, envision them intersecting at a right angle. This kind of relationship can be verified by checking the dot product of their normal vectors. In simple terms:

• The condition for two planes given by \(a_1x + b_1y + c_1z = d_1\) and \(a_2x + b_2y + c_2z = d_2\) to be perpendicular is:
• \(a_1a_2 + b_1b_2 + c_1c_2 = 0\)

For the planes \(x + 2y = 3\) and \(4x + 8y = 5\), substituting the values, we get \(1 \times 4 + 2 \times 8 + 0 \times 0 = 20\). Since it does not equal zero, these planes are not perpendicular. If the dot product had equalled zero, it would mean the planes intersect at a 90-degree angle, which they do not in this case.

Coefficients of Planes

The coefficients in plane equations serve as the key features determining the relation between planes. They are crucial in analyzing the orientation and position in space. When we talk about the coefficients, we're referring to the numbers in front of the variables in the equations of planes, denoted as \(a, b,\) and \(c\).

These numbers reflect the directions of the normal vectors to the plane:

• In equations like \(ax + by + cz = d\), \(a, b, c\) are components that align with respective vectors in x, y, and z directions.
• The relationship between these coefficients across two plane equations illuminates their spatial relation, whether they are parallel or perpendicular.

In the exercise given, by analyzing the coefficients \((1, 2, 0)\) and \((4, 8, 0)\), students can identify the nature of the relationship. A special emphasis is on understanding and calculating their proportionality for parallelism or summation check for perpendicularity. Remember, these coefficients guide in revealing the fundamental nature of plane intersections or alignments.