Problem 22 - 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 planes \(3 x+y-4 z=3\) and \(-9 x-3 y+12 z=4\) are parallel.

Step 1: Extract normal vectors from the plane equations

The normal vector to the plane is given by the coefficients of x, y, and z in the equation of the plane. For the first plane, this gives you a normal vector \((3, 1, -4)\). For the second plane, the normal vector is \(-9, -3, 12)\).

Step 2: Test for parallel planes

Planes are parallel when one normal vector is a multiple of the other. In this case, the normal vector of the second plane is exactly three times the normal vector of the first plane. So the planes are parallel.

Step 3: Confirm with the perpendicular condition

Perpendicular planes would have normal vectors that yield a zero dot product, but as the two planes have already been found to be parallel in the previous step, testing for perpendicularity isn't necessary. However, for completeness, let's calculate the dot product as well: \(3*-9 + 1*-3 - 4*12 = -27 - 3 - 48 = -78 \neq 0\). So, as expected, the planes are not perpendicular.

Key Concepts

Parallel Planes

In plane geometry, understanding the concept of parallel planes is crucial for spatial reasoning. Parallel planes are always equidistant from each other and never intersect. The test for whether two planes are parallel involves comparing their normal vectors. Each plane in the equation form \(a_1 x + b_1 y + c_1 z = d_1\) has a normal vector extracted from the coefficients \((a_1, b_1, c_1)\).

To determine if two planes are parallel:

• The normal vector of one plane must be a scalar multiple of the other.
• Mathematically, this means there is a constant \(k\) such that \(a_1 = k a_2\), \(b_1 = k b_2\), and \(c_1 = k c_2\).

In the example given, the normal vector for the first plane is \((3, 1, -4)\) and for the second plane is \((-9, -3, 12)\). Since the second vector is precisely three times the first one, these planes are affirmatively parallel. Parallelism in planes signifies a uniform separation in three-dimensional space, much like parallel lines in two-dimensional geometry.

Perpendicular Planes

Perpendicular planes in geometry are planes that intersect such that their intersection forms a right angle. The test for perpendicularity between two planes involves the dot product of their normal vectors.

When two planes are perpendicular:

• Their normal vectors' dot product is zero.
• The dot product for normal vectors \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\) is calculated as \(a_1 a_2 + b_1 b_2 + c_1 c_2\).

If this calculation results in zero, the planes make a right angle at their point of intersection. In our scenario, the normal vectors are \((3, 1, -4)\) and \((-9, -3, 12)\). The dot product is \(-27 - 3 - 48 = -78\), which is clearly non-zero. Therefore, these planes are not perpendicular. This illustrates that while parallel and perpendicular are specific properties of planes, not all plane orientations fall under these categories.

Normal Vectors

Normal vectors are fundamental to understanding the orientation of planes in geometry. They are perpendicular to their respective planes and provide a concise way to describe the plane's direction and its orientation in space. Derived from the plane's equation coefficients, the normal vector \((a, b, c)\) succinctly encodes this geometric information.

The role of normal vectors includes:

• Determining parallelism or perpendicularity of planes by analyzing vector relationships.
• Providing critical insights into the spatial organization and orientation of multiple planes.
• Assisting in calculations involving intersections and angles between planes.

In practical applications, such as computer graphics or physics simulations, understanding normal vectors aids in rendering surfaces and calculating forces. In the given exercise, confirming parallelism by observing that one normal vector is a scalar multiple of the other offers a great example of the simplicity and power normal vectors add to geometric analysis.