Problem 28 - 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 \(a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0\). $$ 2 x-z=1,4 x+y+8 z=10 $$

Step-by-Step Solution

Answer

The given planes are perpendicular.

Step 1: Identify the Coefficients

Given plane equations are \(2x - z = 1\) and \(4x + y + 8z = 10\). For the first plane, the coefficients are \(a_1 = 2, b_1 = 0, c_1 = -1\) and for the second plane, the coefficients are \(a_2 = 4, b_2 = 1, c_2 = 8\).

Step 2: Test for Parallel Planes

The planes will be parallel if \(a_1 = ka_2, b_1 = kb_2\), and \(c_1 = kc_2\) for a nonzero constant k. We can use \(a_1\) and \(a_2\) to calculate the potential k value. Dividing \(a_1\) over \(a_2\), we get \(k = 2/4 = 0.5\). Now using same value of \(k = 0.5\), check if \(b_1 = kb_2\) and \(c_1 = kc_2\). For \(b_1 = kb_2\), we have \(0 = 0.5 * 1\) (0 ≠ 0.5), which doesn't hold true. Hence, the given planes are not parallel.

Step 3: Test for Perpendicular Planes

The planes will be perpendicular if \(a_1 * a_2 + b_1 * b_2 + c_1 * c_2 = 0\). Substituting for \(a_1, a_2, b_1, b_2, c_1, c_2\), we get \(2*4 + 0*1 + (-1)*8\), which equals 0. Since this condition holds true, we can conclude that the given planes are perpendicular.

Key Concepts

Parallel Planes

When dealing with plane equations like the ones given in the exercise, understanding whether two planes are parallel is crucial. Parallel planes are like the train tracks that never meet; they remain precisely the same distance apart, no matter how far you extend them. In mathematical terms, two planes will be parallel if their normal vectors (the vectors perpendicular to the plane) are scalar multiples of each other.
To determine if two planes are parallel:

• Identify the coefficients of each plane equation. For example, for the plane equation \(a_{1} x + b_{1} y + c_{1} z = d_{1}\), the coefficients are \(a_{1}, b_{1}, c_{1}\).
• Compare the ratios of the coefficients of the two planes. The planes are parallel if you can find a non-zero constant \(k\) such that \(a_{1} = k a_{2}, b_{1} = k b_{2}, c_{1} = k c_{2}\).

Parallel planes have matching coefficients when scaled by \(k\). In our provided example, after calculating the ratio of coefficients, we see they don't match for all terms, revealing that the planes are not parallel.

Perpendicular Planes

Understanding perpendicular planes can be a bit like visualizing the ceiling meeting the wall. They intersect at right angles, forming a T-shape in space. This concept is defined by the relationship between the normal vectors of the planes involved.
To check if two planes are perpendicular:

• Examine the dot product of their normal vectors, which are extracted from the coefficients of the plane equations.
• The planes are perpendicular if their dot product equals zero. Mathematically, this is expressed as \(a_1 a_2 + b_1 b_2 + c_1 c_2 = 0\).

For the example provided, when substituting the values into the dot product formula, the result equals zero, confirming that these planes intersect at a right angle. The method of checking this involves simple multiplication and addition.

Coefficients of Planes

Coefficients in plane equations serve as the essential elements of understanding spatial relationships. These coefficients are the numbers attached to the variables in the plane equations and represent components of the normal vector to each plane.
Let's explore the importance of coefficients:

• The coefficients \(a\), \(b\), and \(c\) in the equation \(ax + by + cz = d\) define a vector \(\langle a, b, c \rangle\), called the normal vector.
• This normal vector is crucial because it dictates the orientation of the plane in space. Essentially, it points perpendicularly away from the plane.
• Understanding the normal vector aids in determining whether planes are parallel or perpendicular, based on their interactions with each other's normal vectors.

The coefficients let you examine how planes align or intersect in space, highlighting the geometric beauty of their interactions. With practice, reading coefficients can become intuitive, providing a powerful tool for solving geometric problems.