Problem 28 - Calculus Early Transcendentals

Question

Determine if the following pairs of planes are parallel, orthogonal, or neither parallel nor orthogonal. $$3 x+2 y+2 z=10 \text { and }-6 x-10 y+19 z=10$$

Step-by-Step Solution

Answer

Based on the analysis of their direction vectors, the given planes are orthogonal. This is because the dot product of the direction vectors is zero, indicating that the planes have perpendicular directions.

Step 1: Identify Direction Vectors

From the given plane equations, we can find the direction vectors by looking at the coefficients of x, y, and z. For the first plane: \(3x + 2y + 2z = 10\), the direction vector is \(\begin{pmatrix}3\\2\\2\end{pmatrix}\). For the second plane: \(-6x - 10y + 19z = 10\), the direction vector is \(\begin{pmatrix}-6\\-10\\19\end{pmatrix}\).

Step 2: Check for Parallel Planes

If the direction vectors are proportional, then the planes are parallel. We need to verify if there exists a constant k such that \(\begin{pmatrix}-6\\-10\\19\end{pmatrix}=k*\begin{pmatrix}3\\2\\2\end{pmatrix}\). From the x-coordinates, we can see that \(k = \frac{-6}{3} = -2\). Checking the other coordinates: \(-10 = (-2) * 2 = -4\), which is not true, so the vectors are not proportional, and the planes are not parallel.

Step 3: Check for Orthogonal Planes

If the dot product of the direction vectors is zero, then the planes are orthogonal. So, let's calculate the dot product; $$(3, 2, 2) \cdot (-6, -10, 19) = (3 * (-6)) + (2 * (-10)) + (2 * 19) = -18 - 20 + 38 = 0$$ The dot product is zero, which means the planes are orthogonal.

Step 4: Final Answer

The given planes are orthogonal.

Key Concepts

Direction Vectors

When we are dealing with planes in three-dimensional space, direction vectors play an essential role. They help us understand the orientation of the plane. Direction vectors can be easily extracted from plane equations. For instance, in an equation of the plane like \(ax + by + cz = d\), the direction vector is given as \(\begin{pmatrix}a\b\c\end{pmatrix}\).

This direction vector represents a perpendicular vector to the plane. It is fundamental when analyzing relationships between planes, such as determining if they are parallel, orthogonal, or neither.

• To find the direction vector, simply look at the coefficients of \(x\), \(y\), and \(z\).
• Direction vectors are used to compare orientation and relationship between different planes.

Parallel Planes

Parallel planes have a unique spatial relationship where they never intersect. Essentially, parallel planes have direction vectors that are proportional. In mathematical terms, two direction vectors \(\begin{pmatrix}a_1\b_1\c_1\end{pmatrix}\) and \(\begin{pmatrix}a_2\b_2\c_2\end{pmatrix}\) are proportional if there exists some constant \(k\) such that:

\[\begin{pmatrix}a_2\b_2\c_2\end{pmatrix} = k \cdot \begin{pmatrix}a_1\b_1\c_1\end{pmatrix}\]

In practical terms, this means you can multiply one direction vector by a constant factor to achieve the other. If you cannot find such a constant, the planes are not parallel.

• If two planes have proportional direction vectors, they are parallel.
• Parallel planes can be recognized by comparing their coefficients.

Orthogonal Planes

Orthogonal planes intersect each other at a right angle. To determine if two planes are orthogonal, you need to calculate the dot product of their direction vectors.

The dot product of two vectors \(\begin{pmatrix}a_1\b_1\c_1\end{pmatrix}\) and \(\begin{pmatrix}a_2\b_2\c_2\end{pmatrix}\) is calculated as:

\[a_1 \cdot a_2 + b_1 \cdot b_2 + c_1 \cdot c_2\]

If the result is zero, the vectors are orthogonal, indicating that the planes they define are also orthogonal.

• The dot product gives insight into the angle between the two vectors.
• Orthogonal planes intersect at a 90-degree angle, making them perpendicular.