Section 2.5 - Equations of Lines and Planes

Section 2.5 - Equations of Lines and Planes

Lines in 3D Space

In three-dimensional geometry, lines can be represented using vector and parametric equations. These forms are essential for describing the position and direction of a line in space.

• Vector Equation of a Line: The vector equation for a line in 3D passing through point r0 and parallel to direction vector v is: $\mathbf{r} = \mathbf{r}_0 + t\mathbf{v}$ where t is a real parameter.

• Parametric Equations: If the line passes through point $(x_0, y_0, z_0)$ and is parallel to vector $(p_1, p_2, p_3)$, the parametric equations are: $x = x_0 + p_1 t$ $y = y_0 + p_2 t$ $z = z_0 + p_3 t$

• Direction Numbers: The components $p_1, p_2, p_3$ are called direction numbers of the line.

• Symmetric Equations: By eliminating the parameter t, the symmetric equations of the line are: $\frac{x - x_0}{p_1} = \frac{y - y_0}{p_2} = \frac{z - z_0}{p_3}$

• Skew Lines: Two lines are skew if they do not intersect and are not parallel, meaning they do not lie in the same plane.

Example: Find a vector equation and parametric equations for the line passing through the point (5, 1, 3) and parallel to the vector $\langle 1, 4, -2 \rangle$. Verify if the point (6, 5, 1) is on the line.

Planes in 3D Space

Planes in three-dimensional space can be described using vector and scalar equations. The normal vector to the plane is crucial in these representations.

• Vector Equation of a Plane: The vector equation for a plane is: $\mathbf{n} \cdot (\mathbf{r} - \mathbf{r}_0) = 0$ where $\mathbf{n}$ is the normal vector, $\mathbf{r}_0$ is a point on the plane, and $\mathbf{r}$ is a general point on the plane.

• Scalar Equation of a Plane: For a plane passing through $(x_0, y_0, z_0)$ with normal vector $\mathbf{n} = (n_1, n_2, n_3)$: $n_1(x - x_0) + n_2(y - y_0) + n_3(z - z_0) = 0$

Example: Find an equation of the plane through the point (2, 4, 1) with normal vector $\mathbf{n} = (2, 3, 4)$. Find the intercepts and sketch the plane.

Intersection of Lines and Planes

To find the intersection point of a line and a plane, substitute the parametric equations of the line into the plane's equation and solve for the parameter.

• Example: Find the point at which the line with parametric equations $x = 2 + 3t$, $y = -4t$, $z = 5 + t$ intersects the plane $4x + 5y - 2z = 18$.

Parallel and Intersecting Planes

Planes can be parallel or intersecting depending on their normal vectors. The angle between two planes is determined by the angle between their normal vectors.

• Parallel Planes: Two planes are parallel if their normal vectors are parallel.

• Intersecting Planes: If normal vectors are not parallel, the planes intersect in a line. The angle $\theta$ between two planes is given by: $\theta = \arccos\left(\frac{\mathbf{n}_1 \cdot \mathbf{n}_2}{|\mathbf{n}_1||\mathbf{n}_2|}\right)$

• Example: Find the angle between the planes $x + y + z = 1$ and $-2y + 3z = 1$, and find symmetric equations for their line of intersection.

Distance Between Parallel Planes

The distance between two parallel planes can be calculated using their equations.

• Formula: For planes $ax + by + cz = d_1$ and $ax + by + cz = d_2$, the distance $D$ is: $D = \frac{|d_1 - d_2|}{\sqrt{a^2 + b^2 + c^2}}$

• Example: Find the distance between the parallel planes $10x + 2y - 2z = 5$ and $5x + y - z = 1$.

Summary Table: Equations in 3D Geometry

Object	Vector Equation	Parametric/Symmetric Equation	Key Parameters
Line	$\mathbf{r} = \mathbf{r}_0 + t\mathbf{v}$	$x = x_0 + p_1 t$, $y = y_0 + p_2 t$, $z = z_0 + p_3 t$ $\frac{x - x_0}{p_1} = \frac{y - y_0}{p_2} = \frac{z - z_0}{p_3}$	Point $(x_0, y_0, z_0)$, Direction $(p_1, p_2, p_3)$
Plane	$\mathbf{n} \cdot (\mathbf{r} - \mathbf{r}_0) = 0$	$n_1(x - x_0) + n_2(y - y_0) + n_3(z - z_0) = 0$	Point $(x_0, y_0, z_0)$, Normal $(n_1, n_2, n_3)$

Additional info: The notes include several worked examples and diagrams illustrating lines and planes in 3D, as well as formulas for intersection, angles, and distances. These concepts are foundational for multivariable calculus and analytic geometry.