Problem 24 - College Algebra and 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 \(a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0\). $$ x+3 y+2 z=6,4 x-12 y+8 z=24 $$

Step-by-Step Solution

Answer

The given planes are neither parallel nor perpendicular.

Step 1: Identify coefficients of the planes

The coefficients of x, y and z are as follows: For the first plane, \(a_1 = 1\), \(b_1 = 3\), \(c_1 = 2\). For the second plane, \(a_2 = 4\), \(b_2 = -12\), \(c_2 = 8\).

Step 2: Check if the planes are parallel

To check if the two planes are parallel, compare the ratios of the coefficients. If \(a_1/a_2 = b_1/b_2 = c_1/c_2\), then the planes are parallel. Here, \(1/4\), \(3/-12\), and \(2/8\) do not give the same value so planes are not parallel.

Step 3: Check if the planes are perpendicular

To check if the two planes are perpendicular, calculate the dot product of the coefficients. If \(a_1a_2 + b_1b_2 + c_1c_2 = 0\), then the planes are perpendicular. Here, \(1*4 + 3*-12 + 2*8\) = -8, which is not equal to 0. Therefore, the planes are not perpendicular.

Step 4: Conclusion

Based on the computations, the planes are neither parallel nor perpendicular

Key Concepts

Parallel Planes

In the world of geometry, parallel planes are planes that never intersect. They are always the same distance apart, no matter how far you extend them in space.
To determine if two planes are parallel, you must look at their normal vectors. The normal vector is a vector that points perpendicularly away from the plane.
In our original exercise, the equations of the planes given were:

• First Plane: \(a_1x + b_1y + c_1z = d_1\)
• Second Plane: \(a_2x + b_2y + c_2z = d_2\)

The condition for two planes to be parallel is that their normal vectors are scalar multiples of each other. This means there is a constant \(k\) such that:

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

If this condition is met, the two planes will neither meet nor deviate from one another. In the example given, however, the ratios are not equal, so the planes are not parallel.

Perpendicular Planes

Perpendicular planes intersect at a right angle. This geometric property is very useful in construction and design, where forming right angles is often required.
To check whether two planes are perpendicular to each other, calculate the dot product of their normal vectors. For normal vectors \((a_1, b_1, c_1) \text{ and } (a_2, b_2, c_2) \), the dot product is:

• \(a_1a_2 + b_1b_2 + c_1c_2\)

If the dot product equals zero, the planes are perpendicular.
In the exercise, the dot product calculation was:

• \(1\times4 + 3\times(-12) + 2\times8 = -8\)

Since -8 is not zero, the planes are not perpendicular.

Dot Product

The dot product is a fundamental concept in vector algebra, often used to determine the angle between two vectors.
It is calculated as the sum of the products of the corresponding components of two vectors. Mathematically, for vectors \( \mathbf{v} = a_1 \hat{i} + b_1 \hat{j} + c_1 \hat{k} \) and \( \mathbf{w} = a_2 \hat{i} + b_2 \hat{j} + c_2 \hat{k} \), the dot product is represented as:

• \( a_1a_2 + b_1b_2 + c_1c_2 \)

This value gives insight into the geometric relation of the two vectors.

• If the dot product is positive, the angle between the vectors is acute.
• If it is zero, the vectors are perpendicular.
• If negative, the vectors form an obtuse angle.

Understanding the dot product can reveal much about the spatial orientation between two planes, as seen in determining whether they are perpendicular in the provided exercise.