Problem 36 - Precalculus Mathematics for Calculus

Question

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 \(\mathbf{n}\) (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

Line 2 is parallel to the plane, and Line 1 is perpendicular to the plane.

Step 1: Understanding parallel and perpendicular vectors

Two vectors are parallel if one is a scalar multiple of the other, meaning they have the same or exact opposite direction. They are perpendicular if their dot product is zero, indicating they are at right angles to each other.

Step 2: Answering part (a)

If the line is perpendicular to the plane, then the direction vector of the line \( \mathbf{v} \) is parallel to the normal vector \( \mathbf{n} \) of the plane since a normal vector defines a direction perpendicular to the plane itself. Therefore, \( \mathbf{v} \) and \( \mathbf{n} \) must be parallel.

Step 3: Answering part (b)

If the line is parallel to the plane, then its direction vector \( \mathbf{v} \) must be perpendicular to the normal vector \( \mathbf{n} \) of the plane, because a plane's normal vector is by definition perpendicular to every direction that lies within that plane. So \( \mathbf{v} \) and \( \mathbf{n} \) are perpendicular.

Step 4: Finding direction vectors of given lines

The direction vector for Line 1 is \( \mathbf{v}_1 = \langle 2, -2, 8 \rangle \) and for Line 2 is \( \mathbf{v}_2 = \langle -2, 2, 1 \rangle \).

Step 5: Obtaining the plane's normal vector

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

Step 6: Checking which line is parallel to the plane

Line 1 is parallel to the plane if its direction vector \( \mathbf{v}_1 \) is perpendicular to the normal vector \( \mathbf{n} \). The dot product \( \mathbf{v}_1 \cdot \mathbf{n} = 2 \times 1 + (-2) \times (-1) + 8 \times 4 = 2 + 2 + 32 = 36 \), which is not zero, so Line 1 is not parallel to the plane.

Step 7: Testing Line 2 with the normal vector

Line 2 is parallel to the plane if its direction vector \( \mathbf{v}_2 \) is perpendicular to the normal vector \( \mathbf{n} \). The dot product \( \mathbf{v}_2 \cdot \mathbf{n} = (-2) \times 1 + 2 \times (-1) + 1 \times 4 = -2 - 2 + 4 = 0 \), which is zero, indicating Line 2 is parallel to the plane.

Step 8: Checking which line is perpendicular to the plane

Line 1 is perpendicular to the plane since we established it is not parallel, and its direction \( \mathbf{v}_1 \) is quite likely parallel to \( \mathbf{n} \) since the dot product check failed the perpendicular condition with only Line 2 honoring it.

Key Concepts

Perpendicular Vectors

Perpendicular vectors are two vectors that intersect at a right angle, meaning they meet at 90 degrees. This relationship is crucial in understanding how vectors interact in three-dimensional space. Mathematically, the key to identifying whether two vectors are perpendicular is their dot product.

• If the dot product of two vectors is zero, they are perpendicular.
• The dot product is calculated as the sum of the products of their respective components.

In our exercise, if the line, represented by vector \( \mathbf{v} \), is parallel to the plane, it must be perpendicular to the plane's normal vector \( \mathbf{n} \). This scenario implies their dot product is zero, aligning with the concept of perpendicular vectors.

Parallel Vectors

Parallel vectors point in the same or exact opposite directions. They may vary in magnitude, meaning one vector could be a scaled version of the other. Understanding this concept is vital when analyzing problems involving vectors and planes.

• Two vectors \( \mathbf{a} \) and \( \mathbf{b} \) are parallel if there exists a scalar \( c \) such that \( \mathbf{a} = c \mathbf{b} \).
• Parallel vectors do not cross at any point, assuming they lie on the same line or direction.

In the exercise, if the line \( \mathbf{v} \) is perpendicular to the plane, this means that \( \mathbf{v} \) is parallel to the plane's normal vector \( \mathbf{n} \). This understanding helps determine spatial relationships and directions in vector geometry.

Normal Vector

A normal vector is a vector that is perpendicular to a surface or a plane. It is an essential concept in vector math and physics, often used to define the orientation of a plane.

• The normal vector \( \mathbf{n} \) of a plane characterized by an equation \( ax + by + cz = d \) is \( \langle a, b, c \rangle \).
• This vector provides a standard direction that is orthogonal to every line lying within the plane.

In our problem, the normal vector to the given plane \( x - y + 4z = 6 \) is \( \langle 1, -1, 4 \rangle \). This vector is crucial for understanding both perpendicular and parallel relationships between lines and the plane.

Dot Product

The dot product, also known as the scalar product, measures the cosine of the angle between two vectors, providing insight into their orientation relative to each other.

• The dot product of vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \) is calculated as \( \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 \).
• If the dot product is zero, the vectors are perpendicular.
• If the dot product is positive, the vectors point in the same general direction.
• If the dot product is negative, they point in opposite directions.

In the exercise, the direction vector \( \mathbf{v}_2 \) is found to be perpendicular to the normal vector \( \mathbf{n} \) because their dot product results in zero, confirming that line 2 is parallel to the plane. This calculation is a practical application of the dot product concept.