Problem 22 - 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 \(a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0\). $$ 3 x+y-4 z=3,-9 x-3 y+12 z=4 $$

Step-by-Step Solution

Answer

The given planes are parallel and not perpendicular.

Step 1: Identify coefficients and constant terms

The first plane has the equation \(3x + y - 4z = 3\), from which we identify the coefficients \(a_1 = 3, b_1 = 1, c_1 = -4\), and \(d_1 = 3\). The second plane equation \(-9x - 3y + 12z = 4\) gives us \(a_2 = -9, b_2 = -3, c_2 = 12\), and \(d_2 = 4\).

Step 2: Check for parallelism or perpendicularity

To check if the planes are parallel, we look for a constant \(k\) such that \(a_1 = k * a_2, b_1 = k * b_2,\) and \(c_1 = k * c_2\). Dividing \(a_2\) by \(a_1\), \(b_2\) by \(b_1\), and \(c_2\) by \(c_1\), we receive \(-3, -3,\) and \(-3,\) respectively. Hence, \(k = -3\). The planes are therefore parallel. To confirm, check for perpendicularity. This requires that \(a_1 * a_2 + b_1 * b_2 + c_1 * c_2 = 0\), Evaluating \(3 * -9 + 1 * - 3 + -4 * 12\), we get \(-27 - 3 - 48 = -78\), not zero. Thus, the planes are not perpendicular.

Key Concepts

Parallel Planes

Parallel planes are fascinating in geometry. When two planes in 3D space are parallel, they never intersect. Think of them like two endless sheets of paper, sliding along without ever touching each other. To determine if two planes are parallel, compare their coefficients. For the planes given by the equations \(3x + y - 4z = 3\) and \(-9x - 3y + 12z = 4\), you take the coefficients of each term:

• First plane: \(a_1 = 3\), \(b_1 = 1\), \(c_1 = -4\)
• Second plane: \(a_2 = -9\), \(b_2 = -3\), \(c_2 = 12\)

If there is a nonzero constant \(k\) such that each pair of corresponding coefficients maintains the ratio \(a_1 = k a_2\), \(b_1 = k b_2\), and \(c_1 = k c_2\), the planes are parallel.
Here, by dividing the coefficients of the second plane by those of the first, \(-9/3 = -3\), \(-3/1 = -3\), \(12/(-4) = -3\), you find \(k = -3\).
This consistent ratio confirms the planes are parallel.

Perpendicular Planes

Understanding perpendicular planes involves a different approach. Rather than ratios, we consider the dot product of the normal vectors of the planes.
Normal vectors are derived from the coefficients of the planes. For the plane \(3x + y - 4z = 3\), the normal vector is \((3, 1, -4)\). For \(-9x - 3y + 12z = 4\), it is \((-9, -3, 12)\).
To check if the planes are perpendicular, calculate the dot product: \[a_1a_2 + b_1b_2 + c_1c_2\].
If the dot product equals zero, the planes are perpendicular. Using the given planes’ vectors, calculate: \(3(-9) + 1(-3) + (-4)(12) = -27 - 3 - 48 = -78\).
This is not zero, so these planes are not perpendicular. No right angles here!

Coefficients Comparison

Comparing coefficients is a fundamental skill in determining the relationship between two planes. It's all about numbers and how they relate to each other. The coefficients refer to the numbers multiplying the variables \(x\), \(y\), and \(z\) in the equation of a plane.
For example, in the plane equation \(3x + y - 4z = 3\), the coefficients are \(3, 1,\) and \(-4\). By comparing these with another plane’s coefficients, we figure out relationships like parallelism or if nothing special occurs (neither parallel nor perpendicular).

• Check for a consistent ratio (a constant \(k\)) for parallelism.
• Check the sum of products of corresponding coefficients for perpendicularity.

Thus, coefficients help us quickly decide if two planes relate in a simple, predictable way within the space. Once mastered, this comparison becomes a quick mental math exercise!