Vector and Parametric Equations of Lines and Planes in 2-Space and 3-Space

Review of Early High School Linear Equations

• Several forms of a line's equation are established in early secondary education, primarily for two-dimensional space:     - Slope-intercept form: $y = mx + b$, where $m$ represents the slope of the line and the point $(0, b)$ represents the y-intercept.     - Slope-point form: $y = m(x - p) + q$, where $m$ represents the slope of the line and $(p, q)$ represents any specific point located on the line.     - Cartesian form (Standard/General form): $Ax + By + C = 0$. When this equation is rearranged into slope-intercept form, the slope is found to be $m = -\frac{A}{B}$.
• Limitations of these forms:     - In all the above cases, a line is defined as the set of all points $(x, y)$ satisfying the given equation.     - None of these forms are capable of describing a line in three-dimensional space ($3$-space) because the concept of slope only involves two quantities: rise and run.     - To extend the description of a line to $3$-space, a new method must be developed using a starting point and a vector.

Vector and Parametric Equations of Lines in 2-Space

• Fundamental Concept: A line can be generated by identifying a starting point and adding infinite scalar multiples of a specific vector, known as the direction vector.
• Vector Equation in 2-Space:     - $(x, y) = (x_0, y_0) + t(m_1, m_2)$, where $t \in \mathbb{R}$.     - $(x_0, y_0)$ is any point on the line.     - $\vec{m} = (m_1, m_2)$ is the direction vector.
• Parametric Equations in 2-Space:     - These are derived by creating a separate equation for each component of the vector equation:         - $x = x_0 + tm_1$         - $y = y_0 + tm_2$         - where $t \in \mathbb{R}$.
• Uniqueness:     - Neither the point nor the direction vector is unique for a given line.     - Consequently, there are an infinite number of vector and parametric equations that can represent a single line.
• Examples:     - Ex. a): Determine the vector and parametric equations of a line passing through point $A(2, -5)$ with direction vector $\vec{m} = (1, 4)$.     - Ex. b): Sketch the line described in Ex a.     - Ex. (Two points):         - a) Determine vector and parametric equations for the line containing the two points $B(-3, 5)$ and $C(6, -1)$.         - b) Determine the point at which the line intersects the x-axis.         - c) Determine the coordinates of another point that is on the line.         - d) Determine if $P(60, -37)$ is on the line.     - Ex. (Perpendicularity): Determine a vector equation for the line that is perpendicular to $(x, y) = (2, 3) + t(6, -5)$, $t \in \mathbb{R}$, and passes through the point $P(-4, 1)$. Use integer components for the direction vector.

Cartesian Equation of a Line

• Direction Vectors and Slope:     - For a line segment $AB$ with a run of $a$ and a rise of $b$, the slope is $m = \frac{b}{a}$.     - The vector $\vec{m} = (a, b)$ describes the direction of this line or any parallel line, with no restriction on direction numbers $a$ and $b$.     - If a direction vector is $\vec{m} = (a, b)$, this corresponds to a slope of $m = \frac{b}{a}$ except when $a = 0$ (which indicates a vertical line).
• Normal Vector:     - A normal (vector) is defined as a vector that is perpendicular to any vector on a given line.
• Cartesian (Scalar) Equation Equations:     - $Ax + By + C = 0$     - In this form, $\vec{n} = (A, B)$ is a normal vector to the line.
• Relationship conversions:     - Ex: Determine the equivalent vector and parametric equations for the line $y = \frac{3}{4}x - 3$.     - Ex: Determine the equivalent slope-intercept form equation for the line $\vec{r} = (-3, 2) + t(1, 6)$.
• Properties of Normals:     - Parallel or Equal Lines: Two lines are parallel or equal if and only if their normal vectors are scalar multiples of each other ($\vec{n}_1 = k\vec{n}_2$).     - Perpendicular Lines: Two lines are perpendicular if and only if the dot product of their normal vectors is equal to zero ($\vec{n}_1 \cdot \vec{n}_2 = 0$).     - Note: These principles also apply if "normal" is replaced with "direction vectors."
• Further Examples:     - Ex: Determine the Cartesian equation of a line passing through point $A(7, 4)$, which has the normal $\vec{n} = (-2, 5)$.     - Ex: Determine the Cartesian equation of the line that passes through the point $(1, 3)$ and is perpendicular to the line with normal vector $(3, 7)$.     - Ex: Determine the vector and parametric equations for the line $4x - 11y = 13$.

Equations of Lines in 3-Space

• Vector Equation in 3-Space:     - $(x, y, z) = (x_0, y_0, z_0) + t(m_1, m_2, m_3)$, where $t \in \mathbb{R}$.     - $(x_0, y_0, z_0)$ is a point on the line; $\vec{m} = (m_1, m_2, m_3)$ is the direction vector.
• Parametric Equations in 3-Space:     - $x = x_0 + tm_1$     - $y = y_0 + tm_2$     - $z = z_0 + tm_3$     - where $t \in \mathbb{R}$.
• Symmetric Equations:     - Derived by solving each parametric equation for the parameter $t$ and setting them equal to each other:     - $\frac{x - x_0}{m_1} = \frac{y - y_0}{m_2} = \frac{z - z_0}{m_3}$
• Non-uniqueness: similar to 2-space, point and direction vector choice is not unique, leading to infinite valid equations for one line.
• Examples:     - Ex: Determine the vector and parametric equations of the line passing through $P(3, -2, 5)$ and $Q(-1, 4, -6)$.     - Ex: Determine the symmetric equation of the line in the previous example.     - Ex: Determine the parametric and symmetric equations of the line that passes through the points $A(4, 7, 2)$ and $B(15, -7, 11)$.

Vector and Parametric Equations of a Plane

• Concept: Unlike a line (which requires one direction vector), a plane requires two non-collinear direction vectors to define its orientation in space.
• Vector Equation of a Plane:     - $(x, y, z) = (x_0, y_0, z_0) + s(a_1, a_2, a_3) + t(b_1, b_2, b_3)$, where $s, t \in \mathbb{R}$.
• Parametric Equations of a Plane:     - $x = x_0 + sa_1 + tb_1$     - $y = y_0 + sa_2 + tb_2$     - $z = z_0 + sa_3 + tb_3$     - where $s, t \in \mathbb{R}$.
• Note: The starting point $(x_0, y_0, z_0)$ and the two direction vectors $\vec{a}$ and $\vec{b}$ are not unique.
• Examples:     - Ex: Determine vector and parametric equations for the plane that contains the points $A(5, 2, 1)$, $B(-2, 3, -1)$, and $C(4, 0, 3)$.     - Ex: Determine a vector equation of the plane containing the line $\vec{r} = (-1, 4, 2) + s(2, 7, 1)$, $s \in \mathbb{R}$ and the point $P(3, 2, 5)$.     - Ex: A plane $\pi$ has $(x, y, z) = (5, 1, -3) + s(2, 0, 4) + t(-1, 2, 5)$, $s, t \in \mathbb{R}$ as its equation. Determine the point of intersection between $\pi$ and the y-axis.     - Ex: Determine the vector equation for a line that is perpendicular to the plane defined by $(x, y, z) = (4, 18, -5) + s(3, -5, 4) + t(-1, 6, 8)$, $s, t \in \mathbb{R}$ and passes through the point $(12, 0, -3)$.
• Perpendicular Planes Note: To find a perpendicular plane instead of a line, the second direction vector can be anything as long as it is not collinear with the first direction vector found. Using one of the original direction vectors from the initial plane is a common practice.

Cartesian Equation of a Plane

• Derivation:     - Consider a plane with a normal vector $\vec{n} = (A, B, C)$ containing a fixed point $A(x_0, y_0, z_0)$.     - For any arbitrary point $P(x, y, z)$ on the plane, the vector $\vec{AP} = (x - x_0, y - y_0, z - z_0)$ must be perpendicular to the normal vector $\vec{n}$.     - Therefore: $\vec{n} \cdot \vec{AP} = 0$     - $(A, B, C) \cdot (x - x_0, y - y_0, z - z_0) = 0$
• General Cartesian Equation:     - $Ax + By + Cz + D = 0$, where $\vec{n} = (A, B, C)$ is the normal vector to the plane.     - The Cartesian equation is not unique as any scalar multiple of the normal vector can be used.
• Examples:     - Ex: Determine the Cartesian equation of the plane that has normal $\vec{n} = (2, 3, 4)$ and contains the point $A(1, 7, 3)$.     - Ex: Determine the Cartesian equation of the plane containing the points $A(1, 5, 2)$, $B(4, 3, -2)$, and $C(6, -3, -2)$.

Parallel and Perpendicular Planes

• Perpendicular Planes:     - Two planes $\pi_1$ and $\pi_2$ with normals $\vec{n}_1$ and $\vec{n}_2$ are perpendicular if their normals are perpendicular.     - Condition: $\vec{n}_1 \cdot \vec{n}_2 = 0$.
• Parallel Planes:     - Two planes $\pi_1$ and $\pi_2$ with normals $\vec{n}_1$ and $\vec{n}_2$ are parallel if their normals are parallel.     - Condition: $\vec{n}_1 = k\vec{n}_2$ for all non-zero real numbers $k$.
• Example:     - Ex: Show that the planes $\pi_1: 3x + 2y - 4z + 1 = 0$ and $\pi_2: 4x - 2y + 2z + 6 = 0$ are perpendicular.

Angle Between Planes

• When two planes meet, they form two supplementary angles, $\theta_1$ and $\theta_2$, such that $\theta_1 + \theta_2 = 180^{\circ}$.
• Relation to Normal Vectors:     - Consider the quadrilateral formed by the two planes and their two normals.     - The sum of interior angles is $360^{\circ}$.     - Two of the angles are $90^{\circ}$ (the normals are perpendicular to the planes).     - Let $\phi$ be the angle between the normals. Then $\theta_1 + 90^{\circ} + 90^{\circ} + \phi = 360^{\circ}$, which simplifies to $\theta_1 + \phi = 180^{\circ}$.     - Since we also know $\theta_1 + \theta_2 = 180^{\circ}$, it follows that $\phi = \theta_2$.     - Thus, the angle between the normals is equal to one of the angles formed by the planes.
• Calculation:     - Use the dot product formula: $\vec{u} \cdot \vec{v} = |\vec{u}||\vec{v}| \cos(\theta)$.
• Example:     - Ex: Determine the angle between the planes $\pi_1: 5x + 1y - 2z + 4 = 0$ and $\pi_2: 4x - 2y + 2z + 6 = 0$.

Sketching Planes from the Cartesian Equation

• Intercept Method:     - Similar to graphing lines in 2D using intercepts, assign two variables equal to zero to find the intercept of the third variable.     - Example: Graph $3x - y + 2z + 6 = 0$ by finding x-int, y-int, and z-int.
• Case: Origin Pass ($D = 0$):     - If the equation is in the form $Ax + By + Cz = 0$, the plane passes through the origin $(0, 0, 0)$.     - To graph, find additional points by inspection (e.g., set one variable to zero and solve for the others).     - Example: Graph $3x + y - 2z = 0$.
• Dimensional Interpretations of $x = 3$:     - In $1$-space: A single point on a number line.     - In $2$-space: A vertical line.     - In $3$-space: A plane parallel to the yz-plane.
• Dimensional Interpretations of $2x + 3y = 12$:     - In $2$-space: A slanted line.     - In $3$-space: A vertical plane created by extending the line into the third dimension (along the z-axis).
• Recall Grade 9 Methods:     - Graphing lines like $2x + 3y = 12$ using slope and y-intercept or both intercepts.     - Horizontal lines: $y = b$.     - Vertical lines: $x = a$.