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

Step-by-Step Solution

Answer

The given planes are perpendicular.

Step 1: Identify the Coefficients

In the first equation, the coefficients of \(x\), \(y\), and \(z\) are \(2\), \(0\), and \(-1\) respectively. Similarly for the second equation, the coefficients of \(x\), \(y\), and \(z\) are \(4\), \(1\), and \(8\) respectively.

Step 2: Check for Parallelism

Planes are parallel if there exists a constant \(k\) such that \(a_1 = k a_2\), \(b_1 = k b_2\), and \(c_1 = k c_2\). So let's compare the coefficients. For \(x\), the ratio is \(4/2 = 2\). But for \(z\), the ratio is \(8/-1 = -8\), which is not equal to the ratio of \(x\). Hence, the planes are not parallel.

Step 3: Check for Perpendicularity

Planes are perpendicular if \(a_1 a_2 + b_1 b_2 + c_1 c_2 = 0\). Let's check if this condition is fulfilled. The sum of products of corresponding coefficients is \(2*4 + 0*1 + -1*8 = 8 - 8 = 0\). Since the sum equals to zero, the planes are perpendicular.

Key Concepts

Planes

Before we dive into finding relationships between planes, let's understand what a plane is in analytic geometry. Picture a plane as a flat, infinitely large sheet of paper stretching endlessly in all directions. In three-dimensional space, a plane can be mathematically described using the equation: \[a x + b y + c z = d\]where \(a\), \(b\), \(c\), and \(d\) are constants. The variables \(x\), \(y\), and \(z\) are coordinate values representing any point lying on that plane. This equation helps in defining both the position and orientation of a plane in the 3D space.

• \(a\), \(b\), and \(c\) are the coefficients of the equation, dictating the plane's orientation.
• \(d\) controls how far the plane is from the origin.

Grasping the basic concept of a plane sets the stage for understanding more complex ideas, like how planes can be parallel or perpendicular to each other.

Parallel Planes

When we talk about parallel planes in geometry, we mean that two planes stay the same distance apart and do not intersect, no matter how far they are extended. For two planes to be parallel, their normal vectors (arising from the coefficients in their equations) have to be proportional. For the planes given by equations:\[a_1 x+b_1 y+c_1 z = d_1\] \[a_2 x+b_2 y+c_2 z = d_2\]there must exist a non-zero constant \(k\) such that:

• \(a_1 = k a_2\)
• \(b_1 = k b_2\)
• \(c_1 = k c_2\)

In simpler terms, if you multiply all coefficients of the first plane by the same number and get the coefficients of the second plane, they are parallel. However, it is important to note that just because two planes are parallel, it does not mean that they are the same - they can be separated by some distance in space.

Perpendicular Planes

Perpendicular planes form an interesting relationship; they meet at a right angle (90 degrees). To identify perpendicular planes, you need to look at the dot product of the normal vectors from their equations. Given the plane equations:\[a_1 x+b_1 y+c_1 z = d_1\] \[a_2 x+b_2 y+c_2 z = d_2\]the dot product of their coefficients (or normal vectors) must be zero:\[a_1 a_2 + b_1 b_2 + c_1 c_2 = 0\]When this condition is satisfied, it means the angle between the planes is 90 degrees, which confirms their perpendicularity. This intersection often simplifies visualizing spatial relationships in geometry, providing a clearer understanding of 3D space distribution.

Coefficients of Plane Equations

The coefficients \(a\), \(b\), and \(c\) in the plane equation are not just arbitrary numbers; they play a crucial role in determining the plane’s orientation in 3D space. Let's explore their significance.

• Each coefficient is part of the normal vector \((a, b, c)\), a vector pointing perpendicularly from the plane.
• The ratios of these coefficients (when compared with another plane) dictate the parallelism, as covered earlier.
• Their product sum helps determine perpendicularity with another plane's normal vector.

Changing any of these coefficients can alter the plane's tilt or angle relative to other planes or axes, similar to how changing one number in a line’s equation alters its slope. Understanding coefficients lets you mold and predict the spatial behavior of planes.