Problem 36 - Essential Calculus Early Transcendentals

Question

\(35-38=\) Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. $$2 x-3 y+4 z=5, \quad x+6 y+4 z=3$$

Step-by-Step Solution

Answer

The planes are perpendicular to each other.

Step 1: Identify the Normal Vectors

The normal vector to a plane in the equation \( ax + by + cz = d \) is given by \( \langle a, b, c \rangle \). For the first plane \( 2x - 3y + 4z = 5 \), the normal vector is \( \langle 2, -3, 4 \rangle \). For the second plane \( x + 6y + 4z = 3 \), the normal vector is \( \langle 1, 6, 4 \rangle \).

Step 2: Check for Parallelism

Two planes are parallel if their normal vectors are scalar multiples of each other. Compare the normal vectors \( \langle 2, -3, 4 \rangle \) and \( \langle 1, 6, 4 \rangle \). There is no scalar \( k \) such that \( 2 = k \cdot 1 \), \(-3 = k \cdot 6 \), and \( 4 = k \cdot 4 \). Thus, the planes are not parallel.

Step 3: Check for Perpendicularity

Two planes are perpendicular if the dot product of their normal vectors is zero. Compute the dot product: \( \langle 2, -3, 4 \rangle \cdot \langle 1, 6, 4 \rangle = 2\times1 + (-3)\times6 + 4\times4 \). Calculate it: \( 2 - 18 + 16 = 0 \). Since the dot product is zero, the planes are perpendicular.

Key Concepts

Normal Vector

A normal vector is a crucial component when dealing with planes in space. It is essentially a vector that is perpendicular to the plane. For any plane described by the equation \( ax + by + cz = d \), the normal vector can be identified as \( \langle a, b, c \rangle \). This implies that understanding the orientation and spatial relationships between planes heavily depends on their normal vectors.
For example, if you have the plane equation \( 2x - 3y + 4z = 5 \), the normal vector would be \( \langle 2, -3, 4 \rangle \). Similarly, for \( x + 6y + 4z = 3 \), the normal vector is \( \langle 1, 6, 4 \rangle \). By examining these vectors, one can infer a lot about the geometric properties of the planes.

• Normal vectors act like arrows pointing away from the plane.
• They play a crucial role in identifying parallels or perpendicularity.

Parallel Planes

Parallel planes do not intersect and are spaced evenly apart, much like parallel lines. This geometric characteristic is identifiable by examining the normal vectors of the planes. Specifically, if two planes are parallel, their normal vectors will be scalar multiples of each other.
Suppose you have two planes with normal vectors \( \langle a_1, b_1, c_1 \rangle \) and \( \langle a_2, b_2, c_2 \rangle \). The planes are parallel if there exists some scalar \( k \) for which each corresponding component of the vectors satisfies \( a_1 = k \cdot a_2, \) \( b_1 = k \cdot b_2, \) and \( c_1 = k \cdot c_2 \).
For example, consider the normal vectors \( \langle 2, -3, 4 \rangle \) and \( \langle 1, 6, 4 \rangle \). There is no scalar \( k \) that can make these vectors equivalent when multiplied by their components:

• \( 2 eq k \cdot 1 \)
• \(-3 eq k \cdot 6 \)
• \(4 = k \cdot 4 \)

This shows that these planes are not parallel.

So, next time you determine if two planes are parallel, remember to check their normal vectors for scalar multiplication.

Perpendicular Planes

Perpendicular planes meet at a right angle (90 degrees) and can be identified by checking the normal vectors. Specifically, if the dot product of the two normal vectors is zero, the planes are perpendicular.
The dot product between two vectors \( \langle a_1, b_1, c_1 \rangle \) and \( \langle a_2, b_2, c_2 \rangle \) is calculated as \( a_1 \times a_2 + b_1 \times b_2 + c_1 \times c_2 \). If this results in 0, the vectors—and thus the planes—are perpendicular.

Consider the previous example with normal vectors \( \langle 2, -3, 4 \rangle \) and \( \langle 1, 6, 4 \rangle \):

• The dot product is \( 2 \times 1 + (-3) \times 6 + 4 \times 4 = 0 \).

Because this computation yields zero, it confirms that the planes are perpendicular in orientation.

Dot Product

The dot product is a vital mathematical operation used to analyze angles between vectors or to determine particular geometric relations such as perpendicularity. Its formula between vectors \( \langle a_1, b_1, c_1 \rangle \) and \( \langle a_2, b_2, c_2 \rangle \) is given by \( a_1 \cdot a_2 + b_1 \cdot b_2 + c_1 \cdot c_2 \).
When the result of the dot product is zero, it indicates that the vectors are perpendicular to each other, which is a key property in calculating perpendicular planes in three-dimensional space.

• It is a quick computational method for determining the orthogonality of vectors.
• The dot product's outcome provides insight into the type of angles formed between vectors—right angles when zero.

For example, the dot product calculation for normal vectors \( \langle 2, -3, 4 \rangle \) and \( \langle 1, 6, 4 \rangle \) as done earlier:
\( 2 \times 1 + (-3) \times 6 + 4 \times 4 = 0 \).
This conclusion tells us that the planes are meeting at a right angle, confirming their perpendicular nature.