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

Step-by-Step Solution

Answer

The given planes \(x + 2y = 3\) and \(4x + 8y = 5\) are parallel and not perpendicular.

Step 1: Identify Coefficients for Comparison

The normal vector to each plane can be formed by the coefficients of x, y, and z. For the first plane \(x + 2y = 3\) or equivalently \(x + 2y - 0z = 3\), the coefficients (a1, b1, c1) are (1, 2, 0). For the second plane \(4x + 8y =5\) or equivalently \(4x + 8y - 0z = 5\), the coefficients (a2, b2, c2) are (4, 8, 0).

Step 2: Check for Parallelism

Two planes are parallel if there exists a nonzero constant \(k\) such that \(a_{1}=k a_{2}\), \(b_{1}=k b_{2}\), and \(c_{1}=k c_{2}\). Comparing the coefficients, we can see that 1=4*k, 2=8*k, and 0=0*k. Solving for k in the first two equations, we get \(k = 1/4\) in both cases. And, for the third equation, 0=0*k holds true for any real number \(k\). Therefore the condition is satisfied and the planes are parallel.

Step 3: Check for Perpendicularity

Even though we have already found that the planes are parallel, for completeness, we can check the condition for perpendicularity as well. Two planes are perpendicular if \(a_{1} a_{2} +b_{1} b_{2}+c_{1} c_{2}=0\). Substituting the coefficients from step 1, we get \(1*4 + 2*8 + 0*0 = 20\) which is not equal to 0, thus concluding that the planes are not perpendicular.

Key Concepts

Plane Equations

Understanding plane equations is the first step when analyzing relationships like parallelism or perpendicularity between planes. A plane equation typically looks like this: \( ax + by + cz = d \). Here, \( a \), \( b \), and \( c \) are coefficients that dictate the orientation of the plane. The equation essentially describes a flat two-dimensional surface in three-dimensional space.
You can think of \( x \), \( y \), and \( z \) as coordinates of any point on the plane, while \( d \) is the constant that shifts the plane along its normal vector. Different values for \( d \) represent parallel planes that lie at different distances from the origin.
In the given exercise, the plane equations were simplified because they did not include a \( z \) term, making them two-dimensional. Recognizing this is crucial before moving on with steps like determining parallelism or perpendicularity with another plane.

Normal Vectors

Normal vectors play a key role in understanding how planes are oriented in space. A normal vector is perpendicular to the plane and gives us a concise representation of its orientation.
For a plane equation \( ax + by + cz = d \), the normal vector is \( \langle a, b, c \rangle \). This means that the direction in which the plane is facing is determined by the coefficients \( a \), \( b \), and \( c \).
If you recall step 1 from the solution, normal vectors for the planes were \( \langle 1, 2, 0 \rangle \) and \( \langle 4, 8, 0 \rangle \). These vectors help us to establish relationships between the planes such as whether they are parallel or perpendicular, just by comparing the vectors directly.

Perpendicular Planes

When considering perpendicular planes, we're looking to see if the planes intersect at a right angle. The condition for perpendicularity between two planes is based on their normal vectors.
Two planes are perpendicular if the dot product of their normal vectors is zero. For vectors \( \langle a_1, b_1, c_1 \rangle \) and \( \langle a_2, b_2, c_2 \rangle \), the dot product is \( a_1a_2 + b_1b_2 + c_1c_2 \).
In our exercise, substituting the values into \( 1*4 + 2*8 + 0*0 \) results in 20, which is not zero. Thus, these planes are not perpendicular, as previously shown in the step-by-step solution.

Coefficients Comparison

Comparing coefficients is a simple and effective method to determine if two planes are parallel. If the ratios of the corresponding coefficients of two plane equations are equal, the planes are parallel.
The condition is expressed as \( a_1 = k a_2 \), \( b_1 = k b_2 \), and \( c_1 = k c_2 \) where \( k \) is a constant. In our exercise, both \( \frac{1}{4} \) and \( \frac{2}{8} \) equal \( \frac{1}{4} \), fulfilling the parallelism condition.
Checking these ratios is straightforward and provides a quick confirmation of how two planes relate to one another in terms of orientation, as can be seen from both the solution method and explanations given.