Problem 36 - Algebra and Trigonometry

Question

Lines and Planes A line is parallel to the vector \(\mathbf{v},\) and a plane has normal vector \(\mathbf{n}\) . (a) If the line is perpendicular to the plane, what is the relationship between \(\mathbf{v}\) and \(\mathbf{n}\) (parallel or perpendicular)? (b) If the line is parallel to the plane (that is, the line and the plane do not intersect), what is the relationship between \(\mathbf{v}\) and \(\mathbf{n}(\text { parallel or perpendicular) } ?\) (c) Parametric equations for two lines are given. Which line is parallel to the plane \(x-y+4 z=6 ?\) Which line is perpendicular to this plane? Line \(1 : \quad x=2 t, \quad y=3-2 t, \quad z=4+8 t\) Line \(2 : \quad x=-2 t, \quad y=5+2 t, \quad z=3+t\)

Step-by-Step Solution

Answer

(a) Parallel; (b) Perpendicular; Line 2 is parallel; No line is perpendicular.

Step 1: Identify Relationship from Perpendicular Condition

If a line is perpendicular to a plane, the direction vector of the line \( \mathbf{v} \) is aligned with the normal vector \( \mathbf{n} \) of the plane. This means \( \mathbf{v} \) and \( \mathbf{n} \) are parallel vectors.

Step 2: Determine Relationship from Parallel Condition

If a line is parallel to a plane, the line's direction vector \( \mathbf{v} \) is perpendicular to the normal vector \( \mathbf{n} \) of the plane. Thus, \( \mathbf{v} \cdot \mathbf{n} = 0 \), meaning \( \mathbf{v} \) and \( \mathbf{n} \) are perpendicular.

Step 3: Express Plane's Normal Vector

For the plane \( x - y + 4z = 6 \), the normal vector \( \mathbf{n} \) is \( \langle 1, -1, 4 \rangle \).

Step 4: Extract Direction Vectors of Lines

For Line 1 given by equations \( x = 2t, \ y = 3 - 2t, \ z = 4 + 8t \), the direction vector is \( \langle 2, -2, 8 \rangle \). For Line 2 given by equations \( x = -2t, \ y = 5 + 2t, \ z = 3 + t \), the direction vector is \( \langle -2, 2, 1 \rangle \).

Step 5: Check Line 1 for Parallelism to Plane

Calculate the dot product \( \mathbf{v}_1 \cdot \mathbf{n} = \langle 2, -2, 8 \rangle \cdot \langle 1, -1, 4 \rangle = 2 \times 1 + (-2) \times (-1) + 8 \times 4 = 2 + 2 + 32 = 36 eq 0 \). Line 1 is neither parallel nor perpendicular to the plane.

Step 6: Check Line 2 for Parallelism to Plane

Calculate the dot product \( \mathbf{v}_2 \cdot \mathbf{n} = \langle -2, 2, 1 \rangle \cdot \langle 1, -1, 4 \rangle = (-2) \times 1 + 2 \times (-1) + 1 \times 4 = -2 - 2 + 4 = 0 \). Line 2 is parallel to the plane.

Step 7: Determine Line 1 for Perpendicularity to Plane

Since Line 2 is parallel to the plane, check the dot product of Line 1's direction vector to confirm it is not zero, which already indicated in Step 5. Consequently, Line 1 is not perpendicular to the plane.

Key Concepts

Parallel and Perpendicular Lines

In geometry, understanding the relationship between vectors and lines is crucial, especially when examining the orientation of lines to planes. If a line is perpendicular to a plane, this means that the line's direction vector is parallel to the plane's normal vector. The normal vector is a vector that is perpendicular to every line in the plane, and thus, when a line is perpendicular to the plane, their direction vectors must be parallel. This can sometimes be confusing, as we typically associate perpendicularity directly, but here, it's the vectors themselves that align.
On the other hand, if a line is parallel to a plane, the relationship shifts. Here, the line's direction vector is actually perpendicular to the plane's normal vector. This might seem counterintuitive at first, but consider that a line running parallel to a surface won't intersect it unless given an additional offset, like a steeper angle. So, the line's direction vector maintains a perpendicular relationship to the normal vector, reflected mathematically by the dot product of these vectors equalling zero.

Parametric Equations of a Line

Parametric equations succinctly describe how a line is traced out in space. By assigning each coordinate a formula based on a parameter (often "t"), it's easy to see the progression of points along a line.
For example, given parametric equations:

• Line 1: \(x = 2t, y = 3 - 2t, z = 4 + 8t\)
• Line 2: \(x = -2t, y = 5 + 2t, z = 3 + t\)

Understanding these equations allows us to derive direction vectors, which are crucial in determining how lines relate to planes. In general, the coefficients of "t" in the parametric equations correspond directly to the components of the line’s direction vector. Recognizing this connection can greatly help when comparing these vectors to a plane's normal vector and understanding how the line behaves relative to that plane.

Normal Vector to a Plane

A normal vector to a plane is a key concept that fundamentally defines the orientation of the plane in space. It acts almost like an arrow pointing directly away from the plane's surface. For instance, consider the plane represented by the equation \(x - y + 4z = 6\). The normal vector here is \(\langle 1, -1, 4 \rangle\), derived from the coefficients of \(x, y,\) and \(z\) in the equation.
Understanding the normal vector allows us to assess whether lines are parallel or perpendicular to the plane. If a line's direction vector is parallel to the normal vector, it indicates the line is perpendicular to the plane. Conversely, if the dot product of a line's direction vector and the normal vector equals zero, the line is parallel to the plane. Becoming comfortable with these interactions provides a deeper insight into spatial relationships in geometry.