2.5 Equations of Lines and Planes in Space

Lines and planes in 3D space are fundamental concepts in Calculus III. They help us understand how objects exist and move in three dimensions, building on our knowledge of 2D geometry.

We'll learn different ways to describe lines and planes mathematically. This includes vector, parametric, and symmetric equations for lines, as well as vector and scalar equations for planes. We'll also explore relationships between these objects and calculate distances.

Lines and Planes in 3D Space

Equations of lines in space

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• Vector equation of a line represents the line using a position vector $\vec{r}$, a point on the line $\vec{r_0}$, a direction vector $\vec{v}$ parallel to the line, and a scalar parameter $t$ that varies to generate all points on the line
  • Example: $\vec{r} = \langle 1, 2, 3 \rangle + t\langle 4, 5, 6 \rangle$
• Parametric equations of a line describe the line using separate equations for $x$, $y$, and $z$ coordinates, each involving a point on the line $(x_0, y_0, z_0)$, components of a direction vector $(a, b, c)$, and the parameter $t$
  • Example: $x = 1 + 4t$, $y = 2 + 5t$, $z = 3 + 6t$
• Symmetric equations of a line express the line using ratios of differences between coordinates of a general point $(x, y, z)$ and a specific point on the line $(x_0, y_0, z_0)$, divided by corresponding components of a direction vector $(a, b, c)$
  • Example: $\frac{x - 1}{4} = \frac{y - 2}{5} = \frac{z - 3}{6}$

Relationships between lines

• Parallel lines have the same direction vector or direction vectors that are scalar multiples of each other
• Skew lines are non-parallel lines that do not intersect in 3D space
• The orthogonal projection of a line onto a plane can be found by projecting two points on the line onto the plane

Distance between point and line

• Formula calculates the shortest distance $d$ from a point $P(x_0, y_0, z_0)$ to a line defined by a position vector $\vec{r_1}$ and direction vector $\vec{v}$
  • Numerator computes the magnitude of the cross product between the vector from $\vec{r_1}$ to $P$ and the direction vector $\vec{v}$
  • Denominator is the magnitude of the direction vector $\vec{v}$
  • Cross product $\times$ finds the perpendicular vector between $(\vec{r_0} - \vec{r_1})$ and $\vec{v}$, and its magnitude divided by $|\vec{v}|$ gives the perpendicular distance

Vector and scalar plane equations

• Vector equation of a plane expresses the plane using a normal vector $\vec{n}$ perpendicular to the plane, a position vector $\vec{r}$ for any point on the plane, and a specific point $\vec{r_0}$ on the plane
  • Dot product $\cdot$ between $\vec{n}$ and $(\vec{r} - \vec{r_0})$ equals zero for all points on the plane
  • Example: $\langle 1, 2, 3 \rangle \cdot (\vec{r} - \langle 4, 5, 6 \rangle) = 0$
• Scalar equation of a plane describes the plane using coefficients $a$, $b$, $c$ from a normal vector and a constant term $d$ determined by a point on the plane
  • Example: $1x + 2y + 3z + 4 = 0$
• Intersecting planes form a line where they meet, which can be determined using linear algebra techniques

Point-to-plane distance in 3D

• Formula finds the shortest distance $d$ from a point $P(x_0, y_0, z_0)$ to a plane $ax + by + cz + d = 0$
  • Numerator evaluates the left side of the plane equation at point $P$ and takes the absolute value
  • Denominator is the magnitude of the normal vector $\langle a, b, c \rangle$
  • Ratio gives the perpendicular distance from the point to the plane

Angles Between Planes

Angle between intersecting planes

• Formula determines the angle $\theta$ between two intersecting planes with normal vectors $\vec{n_1}$ and $\vec{n_2}$
  • Dot product $\vec{n_1} \cdot \vec{n_2}$ in the numerator finds the projection of one normal vector onto the other
  • Denominator multiplies the magnitudes of the two normal vectors
  • Inverse cosine of the ratio gives the angle $\theta$ between the planes
  • Absolute values ensure the angle is between 0 and 90 degrees